#6808138

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

2034 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)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,21)(17,18)(19,22)(20,26)(24,28)(25,27);(2,4)(13,15,18,26,19,27,16,24)(14,20,21,23,22,28,17,25);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,21,22,17)(15,24,27,26)(20,23,28,25);(6,9)(10,12)(13,17,16,22,19,21,18,14)(15,25,24,28,27,23,26,20)

Polycyclic presentation:

2^9|c^2=h,d^2=fi,e^2=hi,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	#72308239, #72308261, #72308923
maximal normal subgroup of	#72308239, #72308261, #72308923

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715334);