#7146136

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7053332);