#3847499

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1777 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)(4,6)(5,8)(9,10,13,12)(11,15,16,14)(19,20)(22,23)(25,26)(27,33)(28,34)(29,37)(30,31)(32,40)(35,36)(38,39);(1,3,6,5)(2,7,4,8)(17,18,21,24)(19,22,20,23)(26,31)(27,34)(32,37)(35,38);(1,2,6,4)(3,8,5,7)(9,10,13,12)(11,14,16,15)(17,19,21,20)(18,22,24,23)(25,27)(26,28)(29,38)(30,34)(31,33)(32,39)(35,40)(36,37);(1,4,6,2)(3,7,5,8)(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,27)(29,39)(30,33)(31,34)(32,38)(35,37)(36,40);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,23,24,22)(25,29)(26,32)(27,35)(28,36)(30,40)(31,37)(33,39)(34,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=fghi,d^2=fi,e^2=fghi,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, 3754695);