#632407

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1906 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 539603);