#6686852

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6594048);