#7064339

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1808 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)(13,14,19,21,17,23,18,24)(15,25,31,33,26,20,28,35)(16,22,32,36,29,38,30,37)(27,39,42,43,40,34,41,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,27)(17,26)(18,28)(19,31)(21,35)(22,34)(23,25)(24,33)(29,40)(30,41)(32,42)(36,44)(37,43)(38,39);(5,7,8,12)(6,9,10,11)(14,21)(15,26)(16,29)(18,19)(20,33)(22,37)(23,24)(25,35)(34,44)(36,38)(39,43)(41,42);(1,3,4,2)(7,12)(9,11)(13,16,17,29)(14,22,23,38)(15,27,26,40)(18,30,19,32)(20,34,25,39)(21,36,24,37)(28,41,31,42)(33,43,35,44);(6,10)(9,11)(14,23)(15,26)(16,29)(21,24)(28,31)(30,32)(34,39)(43,44)

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,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, 6971535);