#516555

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
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

2086 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=g,e^2=fh,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 423751);