#7062942

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=i,e^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=ehi,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, 6970138);