#7006596

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

2221 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)(5,6)(7,9)(8,10)(11,12)(13,14,18,25,19,16,20,23)(15,22,26,28,17,21,24,27);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,19,17)(14,21,16,22)(18,26,20,24)(23,28,25,27);(5,7)(6,9)(8,12)(10,11)(13,16,19,14)(15,21,17,22)(18,25,20,23)(24,27,26,28);(1,3,4,2)(7,12)(9,11)(13,17,19,15)(14,22,16,21)(18,24,20,26)(23,27,25,28);(6,10)(9,11)(13,18,19,20)(14,23,16,25)(15,24,17,26)(21,27,22,28)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,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, 6913792);