#7064558

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

1929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,23,18,24,15,20)(16,22,25,21,17,27,26,28);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,19)(14,20,24,23)(16,26,17,25)(21,22,28,27);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,25)(17,18)(19,26)(20,27)(22,23)(24,28);(1,3,4,2)(7,12)(9,11)(13,17,18,16)(14,22,24,27)(15,25,19,26)(20,28,23,21);(6,10)(9,11)(13,15,18,19)(14,23,24,20)(16,25,17,26)(21,22,28,27)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=hi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6971754);