#1195704

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

1844 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)(5,6,10,7)(8,12,9,11)(15,19)(18,20)(21,22)(23,29)(24,33)(25,35)(26,27)(28,30)(31,36)(32,34);(5,7,10,6)(8,12,9,11)(14,17)(18,20)(22,27)(23,30)(25,33)(31,34);(5,8,10,9)(6,11,7,12)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,34)(25,32)(26,30)(27,29)(31,35)(33,36);(1,2,3,4)(13,15)(14,18)(16,19)(17,20)(21,24)(22,25)(23,31)(26,35)(27,33)(28,36)(29,32)(30,34);(5,9,10,8)(6,12,7,11)(21,25)(22,24)(23,32)(26,33)(27,35)(28,34)(29,31)(30,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=gh,d^2=hi,e^2=fg,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, 1102900);