#4396978

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1497 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4304174);