#6554568

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
prime radical	#25380	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6461764);