#2563472

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fgi,c^2=ghi,e^2=fghi,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, 2470668);