#7054202

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

2137 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)(15,20,18,32)(16,26,28,34)(17,22,31,40)(21,33,37,24)(25,35,30,39)(27,43,29,44)(36,42,38,41);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,19,18)(14,20,23,32)(16,24,28,33)(17,25,31,30)(21,34,37,26)(22,35,40,39)(27,41,29,42)(36,44,38,43);(5,7,8,12)(6,9,10,11)(13,16,19,28)(14,21,23,37)(15,24,18,33)(17,29,31,27)(20,34,32,26)(22,38,40,36)(25,42,30,41)(35,43,39,44);(2,4)(7,12)(9,11)(13,17,19,31)(14,22,23,40)(15,25,18,30)(16,27,28,29)(20,35,32,39)(21,36,37,38)(24,41,33,42)(26,43,34,44);(6,10)(9,11)(13,15,19,18)(14,20,23,32)(16,24,28,33)(17,30,31,25)(21,34,37,26)(22,39,40,35)(27,42,29,41)(36,43,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=i,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, 6961398);