#405667

order	512
nilpotency class	3
isoclinism	isoclinic class of #394275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1028 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:

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

Polycyclic presentation:

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

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, 312863);