#7054646

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

2220 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)(13,14,19,16)(15,22,25,28)(17,21,18,27)(20,23,26,24);(5,6)(7,11)(8,10)(9,12)(13,15,19,25)(14,20,16,26)(17,24,18,23)(21,28,27,22);(2,4)(5,7,8,12)(6,9,10,11)(13,14)(15,20)(16,19)(17,27)(18,21)(22,24)(23,28)(25,26);(7,12)(9,11)(13,16,19,14)(15,22,25,28)(17,27,18,21)(20,23,26,24);(6,10)(9,11)(13,17,19,18)(14,21,16,27)(15,23,25,24)(20,22,26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,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=df,e^b=ehi,f^b=fi,d^c=dhi,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, 6961842);