#1296429

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1762 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)(5,6,10,8)(7,12,9,11)(21,22,26,27)(23,30,29,25)(24,32,34,28)(31,35,36,33);(5,6,10,8)(7,11,9,12)(13,14,17,16)(15,19,20,18)(22,27)(23,29)(28,32)(31,36);(5,7,10,9)(6,11,8,12)(13,15,17,20)(14,18,16,19)(21,23,26,29)(22,25,27,30)(24,31,34,36)(28,35,32,33);(1,2,3,4)(5,8,10,6)(7,12,9,11)(13,16,17,14)(15,19,20,18)(21,24,26,34)(22,28,27,32)(23,31,29,36)(25,35,30,33);(5,9,10,7)(6,12,8,11)(21,25)(22,29)(23,27)(24,33)(26,30)(28,31)(32,36)(34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fgh,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

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, 1203625);