#4757469

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1463 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)(10,13)(11,12)(17,18,20,22)(19,23,24,21)(25,26)(27,30)(28,33)(29,32)(31,35)(34,37)(36,39)(38,40);(9,10,14,13)(11,15,12,16)(17,18,20,22)(19,21,24,23)(26,27)(28,35)(32,34)(36,40);(9,11,14,12)(10,15,13,16)(17,19)(18,21)(20,24)(22,23)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,13,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,15,12,16)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=f,d^2=fh,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, 4664665);