#2109131

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 2016327);