#512965

order	512
nilpotency class	5
isoclinism	isoclinic class of #512832
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

357 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,17)(13,30)(14,33)(15,21)(18,26)(19,27)(20,31)(22,35)(23,34)(24,52)(25,53)(28,45)(29,41)(32,59)(36,49)(37,50)(38,48)(39,55)(40,54)(42,58)(43,56)(44,57)(46,60)(47,70)(51,66)(61,71)(62,69)(63,67)(64,68)(65,72);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,23)(13,24)(14,25)(15,22)(17,34)(18,36)(19,37)(20,38)(21,35)(26,49)(27,50)(28,51)(29,46)(30,52)(31,48)(32,47)(33,53)(39,61)(40,62)(41,60)(42,65)(43,63)(44,64)(45,66)(54,69)(55,71)(56,67)(57,68)(58,72)(59,70);(10,17)(12,26)(13,27)(14,31)(16,34)(18,19)(20,45)(21,41)(23,49)(24,50)(25,48)(28,58)(29,56)(30,39)(32,57)(33,40)(35,60)(36,37)(38,66)(42,59)(43,44)(46,67)(47,68)(51,72)(52,61)(53,62)(54,55)(63,64)(65,70)(69,71);(11,22)(16,35)(23,46)(24,47)(25,48)(34,60)(36,63)(37,64)(38,53)(49,67)(50,68)(51,69)(52,70)(61,65)(62,66)(71,72)

Polycyclic presentation:

2^9|a^2=e,b^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=di,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260454, #72322390, #72322434
maximal normal subgroup of	#72260454, #72322390, #72322434

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420161);