#2233506

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2050 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,15)(13,14)(17,18,21,24)(19,20,23,22)(26,28)(27,31);(1,3,6,5)(2,7,4,8)(9,11)(10,12)(13,16)(14,15)(19,23)(20,22)(25,26,29,28)(27,30,31,32);(3,5)(7,8)(10,13)(11,15)(17,19,21,23)(18,22,24,20)(25,27)(26,30)(28,32)(29,31);(1,4,6,2)(3,7,5,8)(9,12,14,16)(10,15,13,11)(17,20,21,22)(18,23,24,19);(1,5,6,3)(2,7,4,8)(9,13,14,10)(11,16,15,12)(17,18)(19,22)(20,23)(21,24)(25,28,29,26)(27,30,31,32)

Polycyclic presentation:

2^9|a^2=fgi,b^2=gh,c^2=fgi,d^2=h,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, 2140702);