#1346922

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1760 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=g,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1254118);