#2523297

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1712 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=f,d^2=h,e^2=fhi,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, 2430493);