#758330

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=i,e^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 665526);