#7143165

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2096 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)(3,4)(7,11)(10,12)(13,14,19,16)(15,22,26,28)(17,21,18,27)(20,24,23,25);(6,9)(7,11)(13,15)(14,20)(16,23)(17,25)(18,24)(19,26)(21,28)(22,27);(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,19)(17,27)(18,21)(20,26)(22,24)(25,28);(6,9)(10,12)(13,16,19,14)(15,22,26,28)(17,27,18,21)(20,24,23,25);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,16,27)(15,24,26,25)(20,22,23,28)

Polycyclic presentation:

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

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, 7050361);