#4751475

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(6,8)(10,11)(12,16)(19,23)(22,24)(25,26,31,29)(27,34,30,32)(28,36,38,33)(35,39,37,40);(10,11)(14,15)(18,21)(22,24)(25,27,31,30)(26,32,29,34)(28,37,38,35)(33,40,36,39);(9,10,13,11)(12,14,16,15)(17,18,20,21)(19,22,23,24)(25,28,31,38)(26,33,29,36)(27,35,30,37)(32,39,34,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(25,29,31,26)(27,32,30,34)(28,36,38,33)(35,39,37,40);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,19,20,23)(18,22,21,24)(25,30,31,27)(26,32,29,34)(28,35,38,37)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=g,e^2=fghi,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, 4658671);