#474327

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

668 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)(9,10,14,20,15,21,16,22)(11,23,26,34,27,37,32,17)(12,19,31,36,29,18,13,30)(24,38,40,35,28,39,25,33);(2,5)(6,8)(9,11,16,26,15,27,14,32)(10,17,22,23,21,34,20,37)(12,28,31,40,29,24,13,25)(18,35,30,38,19,33,36,39);(3,7)(6,8)(9,12,15,29)(10,18,21,19)(11,24)(13,14,31,16)(17,33)(20,30,22,36)(23,38)(25,32)(26,40)(27,28)(34,35)(37,39);(1,3,4,7)(2,6,5,8)(9,13,14,29,15,31,16,12)(10,19,20,30,21,18,22,36)(11,25,32,24,27,40,26,28)(17,33,37,38,34,35,23,39)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381523);