#2563782

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

2804 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=hi,d^2=f,e^2=hi,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, 2470978);