#3060039

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1888 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=gi,d^2=gh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=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, 2967235);