#577140

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2094 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fh,e^2=gh,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, 484336);