#7054188

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	#42910	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2115 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)(13,14,19,23)(15,24,27,35)(16,21,30,40)(17,22,18,32)(20,33,36,26)(25,41,28,44)(29,38,31,39)(34,42,37,43);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,28)(17,26)(18,33)(19,27)(21,37)(22,35)(23,36)(24,32)(25,30)(29,43)(31,42)(34,40)(38,44)(39,41);(2,4)(7,12)(9,11)(13,16)(14,21)(15,25)(17,31)(18,29)(19,30)(20,34)(22,39)(23,40)(24,41)(26,43)(27,28)(32,38)(33,42)(35,44)(36,37);(6,10)(9,11)(13,17,19,18)(14,22,23,32)(15,26,27,33)(16,29,30,31)(20,35,36,24)(21,38,40,39)(25,42,28,43)(34,41,37,44)

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6961384);