#401143

order	512
nilpotency class	3
isoclinism	isoclinic class of #394055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

1224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 308339);