#577364

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

1874 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)(6,7)(10,11)(13,14,17,18)(15,19,20,16)(21,22)(23,29)(24,26)(25,32)(27,30)(28,34)(31,35)(33,36);(1,2,4,3)(5,6,9,7)(8,10,12,11)(13,15)(14,16)(17,20)(18,19)(22,24)(23,30)(28,32)(31,33);(1,3,4,2)(8,12)(10,11)(13,16,17,19)(14,15,18,20)(21,23)(22,27)(24,29)(25,33)(26,30)(28,36)(31,34)(32,35);(5,7,9,6)(8,11,12,10)(13,14)(15,16)(17,18)(19,20)(21,24,26,22)(23,29,30,27)(25,32,34,28)(31,36,33,35);(5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,19)(18,20)(21,25,26,34)(22,28,24,32)(23,31,30,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=i,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=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, 484560);