#4895643

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1724 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)(3,5)(4,8)(6,7)(10,14)(12,15)(19,23)(22,24)(25,26,28,30)(27,31,32,29);(2,7)(3,4)(10,14)(11,16)(18,21)(22,24)(27,32)(29,31);(1,3,6,4)(2,8,7,5)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,4,6,3)(2,8,7,5)(9,11,13,16)(10,15,14,12)(25,27)(26,29)(28,32)(30,31);(1,5,6,8)(2,4,7,3)(9,12,13,15)(10,16,14,11)(17,19,20,23)(18,22,21,24)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4802839);