#6811773

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,22,26,28)(23,29,30,31)(24,25,33,34)(27,35,36,32);(2,4)(21,23,25,32,26,30,34,35)(22,27,24,29,28,36,33,31);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(23,30)(27,36)(29,31)(32,35);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,25,28,26,33,34,22)(23,31,32,27,30,29,35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=h,d^2=f,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, 6718969);