#456093

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1081 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,26)(13,17)(14,31)(15,20)(16,24)(18,30)(19,29)(22,32)(23,39)(25,37)(27,33)(28,35)(34,40)(36,38);(2,5)(6,8)(9,11,14,24,15,25,29,40)(10,16,19,21,20,34,31,37)(12,27,13,23,28,22,30,38)(17,32,26,39,18,33,35,36);(3,7)(6,8)(9,12)(10,17)(11,22)(13,29)(14,30)(15,28)(16,32)(18,20)(19,35)(21,36)(23,24)(25,27)(26,31)(33,34)(37,39)(38,40);(1,3,4,7)(2,6,5,8)(9,13,29,12,15,30,14,28)(10,18,31,26,20,17,19,35)(11,23,40,27,25,38,24,22)(16,33,37,39,34,32,21,36)

Polycyclic presentation:

2^9|b^2=f,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 363289);