#4747334

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1721 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fg,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=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, 4654530);