#6816250

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42674	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723446);