#4198865

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1588 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=fgh,d^2=gh,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, 4106061);