#936460

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2114 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=i,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 843656);