#2596643

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=i,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, 2503839);