#6554700

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1991 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)(13,14,19,22,18,23,17,21)(15,16,24,26,20,28,25,27);(1,2,4,3)(5,7,10,8)(6,11,9,12)(13,15)(14,16)(17,25)(18,20)(19,24)(21,27)(22,26)(23,28);(1,3,4,2)(5,6,10,9)(7,12,8,11)(13,16,18,28)(14,20,23,15)(17,26,19,27)(21,25,22,24);(5,8,10,7)(6,11,9,12)(13,17,18,19)(14,21,23,22)(15,24,20,25)(16,26,28,27);(5,9,10,6)(7,12,8,11)(13,17,18,19)(14,22,23,21)(15,25,20,24)(16,26,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283353, #72283473, #72284610, …
maximal normal subgroup of	#72283353, #72283473, #72284610, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6461896);