#545247

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

1882 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)(5,6,8,10)(7,9,11,12)(14,18)(16,19)(21,22)(23,26)(24,27)(25,30)(28,33)(29,31)(32,35)(34,36);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,19,20,16)(22,23)(24,31)(28,30)(32,36);(7,11)(9,12)(14,18)(15,20)(21,23,26,22)(24,27,31,29)(25,30,33,28)(32,34,36,35);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,18)(17,20)(21,24,26,31)(22,27,23,29)(25,32,33,36)(28,34,30,35);(5,6)(7,9)(8,10)(11,12)(13,16,17,19)(14,15,18,20)(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=g,d^2=g,e^2=h,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, 452443);