#6467521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,20,18,24,17,23)(15,25,26,36,28,21,29,33)(16,22,30,34,32,37,31,38)(27,39,40,44,41,35,42,43);(2,4)(14,20)(15,26)(16,30)(17,19)(21,25)(22,37)(23,24)(27,41)(28,29)(31,32)(35,44)(39,43);(6,9)(10,12)(13,15,17,29,18,28,19,26)(14,21,23,36,24,25,20,33)(16,27,31,42,32,41,30,40)(22,35,38,44,37,39,34,43);(13,16,17,31,18,32,19,30)(14,22,23,38,24,37,20,34)(15,27,29,42,28,41,26,40)(21,35,36,44,25,39,33,43);(5,7,8,11)(6,10,9,12)(15,28)(21,25)(26,29)(27,41)(33,36)(35,39)(40,42)(43,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=hi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

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, 6374717);