#3399659

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1677 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,20)(21,22)(23,24)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,38)(39,40);(1,3)(2,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,21)(22,24)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,38,40,37);(2,5)(3,8)(9,10)(11,15)(12,14)(13,16)(17,20,22,23)(18,24,21,19)(25,28,31,27)(26,32,34,29)(30,35,39,36)(33,37,40,38);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,20)(21,23)(22,24)(25,29,31,32)(26,27,34,28)(30,38,39,37)(33,36,40,35);(1,5,6,2)(3,7,8,4)(9,11,12,16)(10,13,14,15)(17,21,22,18)(19,20,24,23)(25,30,31,39)(26,33,34,40)(27,35,28,36)(29,37,32,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=fi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3306855);