#589186

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1924 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=fg,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,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, 496382);