#1392147

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2570 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,24)(20,23)(21,22)(27,31)(30,32);(1,3,5,6)(2,4,7,8)(9,11)(10,14)(12,16)(13,15)(17,18,22,21)(19,24,20,23);(9,11)(10,14)(12,16)(13,15)(17,19,22,20)(18,23,21,24)(25,26,29,28)(27,30,31,32);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,20,22,19)(18,23,21,24)(25,27)(26,30)(28,32)(29,31);(1,4)(2,6)(3,7)(5,8)(9,12)(10,15)(11,16)(13,14)(17,21,22,18)(19,24,20,23)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|b^2=fgh,c^2=hi,d^2=h,e^2=hi,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, 1299343);