#690451

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=h,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=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, 597647);