#513013

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

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

(1,2)(3,6)(4,5)(7,8)(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);(2,5)(6,8)(9,11,15,22)(10,16,21,35)(12,23,29,46)(13,24,32,47)(14,25,31,48)(17,34,41,60)(18,36,43,63)(19,37,44,64)(20,38,33,53)(26,49,56,67)(27,50,57,68)(28,51,54,69)(30,52,59,70)(39,61,42,65)(40,62,45,66)(55,71,58,72);(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);(1,3,4,7)(2,6,5,8)(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|b^2=i,d^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, #72322373, #72322380
maximal normal subgroup of	#72260454, #72322373, #72322380

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420209);