#6686953

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2009 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6594149);