#7146138

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

2191 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)(13,14,17,16,18,21,26,27)(15,20,24,22,23,25,28,19);(6,9)(7,11)(13,15,18,23)(14,19,21,22)(16,25,27,20)(17,28,26,24);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,22)(17,24)(18,23)(19,27)(21,25)(26,28);(6,9)(10,12)(13,16,26,14,18,27,17,21)(15,22,28,20,23,19,24,25);(5,7,8,11)(6,10,9,12)(14,21)(15,23)(16,27)(24,28)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,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, 7053334);