#6467634

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#25775	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2271 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=h,e^2=g,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^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374830);