#4416504

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=fi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,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, 4323700);