#1033752

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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

1950 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=g,d^2=f,e^2=h,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, 940948);