#6816143

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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

2506 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283184, #72283215, #72308293
maximal normal subgroup of	#72283184, #72283215, #72308293

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723339);