#550233

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

1984 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)(5,6,9,8)(7,10,12,11)(14,17)(18,20)(21,22,26,24)(23,29,30,27)(25,32,33,28)(31,34,36,35);(1,2,4,3)(5,6,9,8)(7,11,12,10)(14,17)(15,19)(22,24)(23,30)(28,32)(31,36);(1,2,4,3)(5,7,9,12)(6,10,8,11)(13,14,16,17)(15,18,19,20)(21,23,26,30)(22,27,24,29)(25,31,33,36)(28,34,32,35);(5,6,9,8)(7,10,12,11)(13,15,16,19)(14,18,17,20)(21,24,26,22)(23,27,30,29)(25,32,33,28)(31,34,36,35);(1,3,4,2)(5,8,9,6)(7,11,12,10)(21,25,26,33)(22,28,24,32)(23,31,30,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgi,d^2=h,e^2=i,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, 457429);