#7064681

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1829 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=h,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6971877);