#513016

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
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

371 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 114 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,18)(13,17)(14,33)(15,34)(16,23)(19,30)(20,31)(21,29)(24,46)(25,56)(26,38)(27,39)(28,60)(32,45)(35,50)(36,66)(37,68)(40,61)(41,59)(42,57)(43,58)(44,62)(47,53)(48,54)(49,52)(51,63)(55,65)(64,72)(67,71)(69,70);(2,5)(6,8)(9,11)(10,16)(12,23)(13,24)(14,25)(15,26)(17,35)(18,22)(19,36)(20,37)(21,38)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,63)(34,49)(39,46)(40,64)(41,65)(42,47)(43,48)(44,67)(45,69)(56,60)(57,66)(58,68)(59,70)(61,71)(62,72);(10,17)(13,30)(14,31)(15,29)(16,35)(18,39)(19,20)(21,45)(22,46)(24,53)(25,54)(26,52)(27,57)(28,58)(32,62)(33,40)(34,41)(36,37)(38,69)(42,43)(44,60)(47,48)(49,65)(50,66)(51,68)(55,72)(56,67)(59,61)(63,64)(70,71);(1,3,4,7)(2,6,5,8)(11,23)(16,22)(24,50)(25,51)(26,52)(35,46)(36,47)(37,48)(38,49)(53,66)(54,68)(55,70)(56,63)(64,67)(65,69)(71,72)

Polycyclic presentation:

2^9|d^2=ei,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, #72322373
maximal normal subgroup of	#72260454, #72322373

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420212);