#427519

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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1319 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)(4,5)(7,8)(9,10)(11,21)(12,24)(13,16)(14,19)(15,30)(17,26)(18,28)(20,29)(22,31)(23,32)(25,35)(27,34)(33,36)(37,39)(38,40);(9,11)(10,16)(12,25,26,22)(13,14)(15,23)(17,31,24,35)(18,30)(19,21)(20,32)(27,39,40,36)(28,29)(33,34,37,38);(2,5)(6,8)(9,12)(10,17)(11,22)(13,25)(14,26)(15,27)(16,31)(18,33)(19,24)(20,34)(21,35)(23,36)(28,39)(29,40)(30,38)(32,37);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,18,19,32)(12,25,26,22)(15,23,29,28)(16,20,21,30)(17,33,24,37)(27,36,40,39)(31,34,35,38)

Polycyclic presentation:

2^9|b^2=ei,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=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, 334715);