#2411326

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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

1799 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)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(18,21)(19,23)(25,26,29,28)(27,30,32,31);(1,3,6,5)(2,7,4,8)(11,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,27,29,32)(26,30,28,31);(1,2,6,4)(3,8,5,7)(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,22,21,24)(25,27)(26,30)(28,31)(29,32);(1,4,6,2)(3,7,5,8)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,21)(23,24)(25,26,29,28)(27,30,32,31);(1,5,6,3)(2,7,4,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=i,d^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

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, 2318522);