#1199658

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1732 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)(5,6,8,10)(7,11,12,9)(21,22,26,25)(23,27,31,30)(24,32,34,28)(29,36,33,35);(13,14,17,16)(15,19,20,18)(22,25)(27,30)(28,32)(35,36);(5,6)(7,9)(8,10)(11,12)(13,15,17,20)(14,18,16,19)(21,23,26,31)(22,27,25,30)(24,33,34,29)(28,36,32,35);(1,2,3,4)(21,24)(22,28)(23,29)(25,32)(26,34)(27,35)(30,36)(31,33);(5,7,8,12)(6,9,10,11)(13,16,17,14)(15,19,20,18)(21,25,26,22)(23,30,31,27)(24,32,34,28)(29,36,33,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=fh,d^2=fi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1106854);