#6809876

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2461 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)(3,4)(5,6)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(14,18)(15,21)(16,22)(19,24)(20,26)(23,27);(2,4)(7,12)(9,11)(13,14,16,18,17,19,22,24)(15,20,21,25,23,26,27,28);(6,10)(9,11)(14,19)(18,24)(20,26)(25,28);(13,15,22,27,17,23,16,21)(14,20,24,28,19,26,18,25)

Polycyclic presentation:

2^9|c^2=fh,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308561, #72308580, #72308923
maximal normal subgroup of	#72308561, #72308580, #72308923

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717072);