#1388052

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2379 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=hi,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, 1295248);