#1082701

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2020 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=h,d^2=i,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=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, 989897);