#4190651

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

1436 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=fgh,d^2=fgh,e^2=fgh,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, 4097847);