#7006224

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

2142 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)(13,14,18,24,19,23,26,22)(15,25,17,21,27,20,16,28);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,20,23,25)(16,18,17,26)(21,24,28,22);(5,7)(6,9)(8,12)(10,11)(13,16)(14,20)(15,26)(17,19)(18,27)(21,24)(22,28)(23,25);(1,3,4,2)(7,12)(9,11)(13,17,19,16)(14,21,23,28)(15,18,27,26)(20,22,25,24);(6,10)(9,11)(13,18,19,26)(14,22,23,24)(15,16,27,17)(20,28,25,21)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,d^2=f,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, 6913420);