#982859

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1728 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fhi,d^2=f,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 890055);