#1085839

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

2200 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)(7,11,12,9)(13,14,17,15)(16,19,20,18)(21,22,26,27)(23,29,30,24)(25,33,34,28)(31,32,36,35);(7,12)(9,11)(13,15,17,14)(16,19,20,18)(22,27)(23,30)(25,34)(32,35);(5,7,8,12)(6,9,10,11)(13,16)(14,18)(15,19)(17,20)(21,23,26,30)(22,24,27,29)(25,31,34,36)(28,32,33,35);(1,2)(3,4)(21,24)(22,23)(25,32)(26,29)(27,30)(28,31)(33,36)(34,35);(5,6)(7,9)(8,10)(11,12)(13,16)(14,18)(15,19)(17,20)(21,25,26,34)(22,28,27,33)(23,31,30,36)(24,32,29,35)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

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, 993035);