#6810611

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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

1927 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=f,e^2=fh,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	#72308331, #72308367
maximal normal subgroup of	#72308331, #72308367

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717807);