#2177421

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2678 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,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, 2084617);