#3883662

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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

1578 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=g,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3790858);