#7062291

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	#42643	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1865 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)(7,9)(8,10)(11,12)(13,14,18,23,17,22,30,20)(15,24,27,34,25,19,31,32)(16,21,29,36,28,37,41,35)(26,38,40,44,39,33,42,43);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,26)(17,25)(18,27)(20,34)(21,33)(22,24)(23,32)(28,39)(29,40)(30,31)(35,44)(36,43)(37,38)(41,42);(5,7,8,12)(6,9,10,11)(14,20)(15,25)(16,28)(18,30)(19,32)(21,36)(22,23)(24,34)(33,44)(35,37)(38,43)(40,42);(2,4)(7,12)(9,11)(13,16)(14,21)(15,26)(17,28)(18,29)(19,33)(20,35)(22,37)(23,36)(24,38)(25,39)(27,40)(30,41)(31,42)(32,43)(34,44);(6,10)(9,11)(14,22)(16,28)(19,24)(20,23)(26,39)(29,41)(32,34)(40,42)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,g^2=i,b^a=bf,c^a=cg,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, 6969487);