#693163

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

2040 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)(5,6,10,12)(7,8,9,11)(13,14,17,16)(15,18,19,20)(21,22,26,27)(23,29,30,24)(25,28,33,34)(31,35,36,32);(5,7,10,9)(6,8,12,11)(15,19)(18,20)(22,27)(23,30)(25,33)(32,35);(5,8,10,11)(6,7,12,9)(13,15)(14,18)(16,20)(17,19)(21,23,26,30)(22,24,27,29)(25,31,33,36)(28,32,34,35);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,24)(22,23)(25,32)(26,29)(27,30)(28,31)(33,35)(34,36);(5,9,10,7)(6,11,12,8)(13,16,17,14)(15,20,19,18)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=g,e^2=gh,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, 600359);