#422878

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1913 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,d^2=i,e^2=i,g^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,e^c=ei,g^c=gi,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, 330074);