#6686927

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2043 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)(15,18)(16,24)(17,21)(19,22)(20,26)(23,27)(25,28);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15)(14,18)(16,24)(17,22)(19,21)(20,25)(23,27)(26,28);(2,4)(13,16,18,26,19,27,17,25)(14,20,21,24,22,28,15,23);(5,7,9,8)(6,10,11,12)(13,14)(15,17)(16,20)(18,21)(19,22)(23,25)(24,26)(27,28);(2,4)(5,8,9,7)(6,10,11,12)(13,17,19,18)(14,21,22,15)(16,25,27,26)(20,24,28,23)

Polycyclic presentation:

2^9|a^2=f,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,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, 6594123);