#7145913

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2322 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,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, 7053109);