#1614500

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2012 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gh,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1521696);