#582565

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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

1706 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)(7,11)(8,9)(10,12)(13,14,18,16)(15,20,17,19)(22,26)(23,29)(28,33)(30,35);(1,3,4,2)(5,6,8,9)(7,10,12,11)(13,15,18,17)(14,19,16,20)(23,29)(27,32)(30,35)(34,36);(6,9)(10,11)(13,15,18,17)(14,20,16,19)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,20,17,19)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36);(5,7)(6,10)(8,12)(9,11)(13,17,18,15)(14,20,16,19)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 489761);