#4198876

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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

1644 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fghi,c^2=fgh,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4106072);