#2080867

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3022 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=h,d^2=fg,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, 1988063);