#516472

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

2070 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)(6,9)(10,12)(15,19)(18,20)(21,22,26,23)(24,27,32,29)(25,30,33,28)(31,35,36,34);(6,9)(10,12)(13,14,17,16)(15,18,19,20)(22,23)(27,29)(28,30)(34,35);(5,6,8,9)(7,10,11,12)(13,15)(14,18)(16,20)(17,19)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,20,19,18)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=h,d^2=f,e^2=g,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, 423668);