#3855723

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1516 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=fghi,d^2=fhi,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, 3762919);