#545283

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=gh,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, 452479);