#3645596

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1742 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=f,e^2=ghi,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, 3552792);