#550581

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

1872 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)(14,17)(18,20)(21,22,25,26)(23,27,29,30)(24,32,33,28)(31,36,35,34);(1,2,4,3)(6,9)(10,12)(15,19)(18,20)(22,26)(23,29)(28,32)(31,35);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,18,19,20)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(13,15,16,19)(14,18,17,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,36)(33,35);(1,3,4,2)(5,7,8,11)(6,10,9,12)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fghi,d^2=fg,e^2=hi,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 457777);