#553518

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

1926 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=fg,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, 460714);