#7054713

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#25109	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2106 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)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,24,26,20)(16,21,31,40)(17,32,35,23)(19,33,37,27)(25,39,29,38)(28,36,43,44)(30,42,34,41);(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,22,37)(16,28,31,43)(17,33,35,27)(20,32,24,23)(21,39,40,38)(25,30,29,34)(36,41,44,42);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,29)(17,27)(18,26)(19,23)(21,39)(22,24)(25,31)(28,30)(32,37)(33,35)(34,43)(36,41)(38,40)(42,44);(7,12)(9,11)(13,16,18,31)(14,21,22,40)(15,25,26,29)(17,34,35,30)(19,36,37,44)(20,38,24,39)(23,42,32,41)(27,28,33,43);(6,10)(9,11)(14,22)(15,26)(21,40)(23,32)(25,29)(27,33)(28,43)(41,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961909);