#7149173

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

1911 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)(7,11)(10,12)(13,14,19,31,18,22,17,23)(15,24,30,20,27,37,26,34)(16,21,32,42,28,36,29,35)(25,38,41,33,39,44,40,43);(1,2,3,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,15,18,27)(14,20,22,34)(16,25,28,39)(17,30,19,26)(21,33,36,43)(23,37,31,24)(29,41,32,40)(35,44,42,38);(2,4)(6,9)(10,12)(13,16,18,28)(14,21,22,36)(15,25,27,39)(17,29,19,32)(20,33,34,43)(23,35,31,42)(24,38,37,44)(26,40,30,41);(5,7,8,11)(6,10,9,12)(14,22)(16,28)(20,34)(23,31)(24,37)(25,39)(29,32)(40,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7056369);