#1097470

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

1814 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)(5,6,10,12)(7,8,11,9)(21,22)(23,24)(25,33)(26,27)(28,29)(30,34)(31,35)(32,36);(5,7,10,11)(6,8,12,9)(13,14,17,16)(15,19,20,18)(22,27)(23,29)(25,34)(32,35);(5,8,10,9)(6,7,12,11)(13,15,17,20)(14,18,16,19)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,33)(34,36);(1,2)(3,4)(5,9,10,8)(6,11,12,7)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31);(5,6)(7,8)(9,11)(10,12)(13,16,17,14)(15,19,20,18)(21,25,26,34)(22,30,27,33)(23,31,29,36)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=gh,d^2=g,e^2=h,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, 1004666);