#7006218

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

Polycyclic presentation:

2^9|a^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6913414);