#3466416

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2078 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gh,d^2=gi,e^2=fg,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, 3373612);