#398615

order	512
nilpotency class	3
isoclinism	isoclinic class of #394378
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1167 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=efi,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=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, 305811);