#1116937

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

1752 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgh,d^2=f,e^2=g,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, 1024133);