#425818

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1335 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)(3,6)(4,8)(5,7)(9,10)(11,21)(12,25)(13,30)(14,32)(15,20)(16,24)(17,28)(18,31)(19,22)(23,38)(26,37)(27,35)(29,36)(33,39)(34,40);(1,3,5,4)(2,6,7,8)(9,11,22,31,15,13,14,24)(10,16,32,30,20,18,19,21)(12,23,29,40,28,26,27,39)(17,34,36,38,25,33,35,37);(1,4,5,3)(2,6,7,8)(9,12,15,28)(10,17,20,25)(11,23,13,26)(14,29,22,27)(16,33,18,34)(19,36,32,35)(21,37,30,38)(24,40,31,39);(9,13,15,11)(10,18,20,16)(12,26,28,23)(14,24,22,31)(17,34,25,33)(19,21,32,30)(27,39,29,40)(35,38,36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333014);