#6809912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2158 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)(13,14,17,19)(15,20,24,16)(18,21,23,25)(22,26,28,27);(5,7)(6,9)(8,12)(10,11)(14,19)(15,22)(16,26)(18,23)(20,27)(24,28);(2,4)(7,12)(9,11)(13,15,23,28,17,24,18,22)(14,20,25,27,19,16,21,26);(6,10)(9,11)(14,19)(15,24)(21,25)(22,28);(13,16,23,26,17,20,18,27)(14,15,25,28,19,24,21,22)

Polycyclic presentation:

2^9|a^2=i,c^2=fh,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,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717108);