#409986

order	512
nilpotency class	3
isoclinism	isoclinic class of #394038
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317182);