#1168292

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

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, 1075488);