#6809866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806527
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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2246 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)(13,14,19,18)(15,20,25,28)(16,21,17,22)(23,26,24,27);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,26)(17,27)(18,28)(19,25)(21,24)(22,23);(2,4)(7,12)(9,11)(13,16,18,22,19,17,14,21)(15,23,20,26,25,24,28,27);(6,10)(9,11)(13,14,19,18)(15,20,25,28)(16,21,17,22)(23,26,24,27);(13,17,18,21,19,16,14,22)(15,24,20,27,25,23,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308331
maximal normal subgroup of	#72308331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717062);