#1043707

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2268 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=i,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, 950903);