#1499558

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2076 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)(3,7)(4,5)(6,8)(10,13)(11,12)(17,18)(19,21)(20,23)(22,24)(27,32)(28,30);(2,5)(6,7)(9,10)(11,16)(12,15)(13,14)(17,18,20,23)(19,22,24,21)(25,26,29,31)(27,30,32,28);(9,11,14,12)(10,15,13,16)(17,19)(18,21)(20,24)(22,23)(25,27)(26,30)(28,31)(29,32);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,15,13,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,30)(28,31)(29,32);(1,2)(3,7)(4,5)(6,8)(9,13,14,10)(11,16,12,15)(17,18)(19,21)(20,23)(22,24)(25,28,29,30)(26,27,31,32)

Polycyclic presentation:

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

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, 1406754);