#6552514

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1978 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459710);