#4057233

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1854 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)(3,7)(9,10,14,12)(11,13,16,15)(17,18)(19,22)(20,24)(21,23)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,39,38,40);(3,7)(6,8)(9,10,14,12)(11,13,16,15)(17,19,21,24)(18,22,23,20)(26,30)(27,34)(28,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,15,12,13)(17,20)(18,19)(21,22)(23,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,13,16,15)(25,27,29,34)(26,31,30,33)(28,38,37,35)(32,40,36,39);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,28,29,37)(26,32,30,36)(27,35,34,38)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=fh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=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, 3964429);