#572552

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

1946 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)(5,6,8,9)(7,11,12,10)(13,14,18,20)(15,17,16,19)(21,22,27,29)(23,30,24,26)(25,33,35,28)(31,34,32,36);(6,9)(7,12)(13,15)(14,19)(16,18)(17,20)(21,23,27,24)(22,26,29,30)(25,31,35,32)(28,34,33,36);(6,9)(10,11)(14,20)(15,16)(21,24,27,23)(22,26,29,30)(25,31,35,32)(28,36,33,34);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,17,20,19)(21,25,27,35)(22,28,29,33)(23,31,24,32)(26,34,30,36);(1,3,4,2)(5,7,8,12)(6,10,9,11)(13,17)(14,15)(16,20)(18,19)(21,26)(22,23)(24,29)(25,34)(27,30)(28,31)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 479748);