#539468

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

2082 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)(14,17)(15,19)(21,22,26,24)(23,28,31,27)(25,32,34,29)(30,35,33,36);(6,9)(10,12)(14,17)(18,20)(21,23,26,31)(22,27,24,28)(25,33,34,30)(29,36,32,35);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,27)(25,30)(26,31)(29,36)(32,35)(33,34);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,22)(23,28,31,27)(25,32,34,29)(30,36,33,35);(21,25,26,34)(22,29,24,32)(23,30,31,33)(27,35,28,36)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=fghi,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 446664);