#558398

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=fgi,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

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, 465594);