#918326

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2472 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 825522);