#633952

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 541148);