#512386

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

762 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419582);