#6686905

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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

2060 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)(14,18)(15,21)(17,24)(19,26)(22,27)(23,28);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,19)(16,18)(17,20)(21,27)(22,23)(24,25)(26,28);(2,4)(13,15,17,21,16,23,24,28)(14,19,20,27,18,22,25,26);(5,7,9,8)(6,10,11,12)(13,14)(15,22)(16,18)(17,20)(19,23)(21,26)(24,25)(27,28);(2,4)(5,8,9,7)(6,10,11,12)(14,18)(19,22)(20,25)(26,27)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=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, 6594101);