#6807321

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=fh,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714517);