#7006275

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

2216 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)(13,14,20,28,18,16,19,23)(15,24,17,22,25,21,26,27);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,25)(14,21,16,24)(17,19,26,20)(22,28,27,23);(5,7)(6,9)(8,12)(10,11)(13,16,18,14)(15,24,25,21)(17,27,26,22)(19,23,20,28);(1,3,4,2)(7,12)(9,11)(13,17,18,26)(14,22,16,27)(15,20,25,19)(21,23,24,28);(6,10)(9,11)(15,25)(17,26)(21,24)(22,27)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6913471);