#1287249

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

2676 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)(4,7)(5,8)(10,11)(12,16)(18,21)(22,24)(25,26)(27,32)(28,30)(29,31);(1,3)(2,5)(4,8)(6,7)(25,27)(26,28)(29,30)(31,32);(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,32)(28,30)(29,31);(1,3)(2,6)(4,8)(5,7)(9,11,13,10)(12,15,16,14)(17,19,20,23)(18,22,21,24)(25,28,29,32)(26,30,31,27);(1,2)(3,5)(4,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,28)(29,31)(30,32)

Polycyclic presentation:

2^9|c^2=i,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1194445);