#418337

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

1725 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)(3,4)(6,7)(9,10,13,18)(11,21,24,16)(12,26,28,36)(14,20,15,19)(17,30,35,29)(22,31,25,33)(23,38,27,40)(32,37,34,39);(5,6)(7,8)(9,11)(10,16)(12,27)(13,24)(14,22)(15,25)(17,34)(18,21)(19,31)(20,33)(23,28)(26,38)(29,39)(30,37)(32,35)(36,40);(2,4)(11,22)(14,15)(16,31)(19,20)(21,33)(23,37)(24,25)(26,36)(27,39)(29,30)(32,40)(34,38);(5,7,8,6)(9,12)(10,17)(11,23)(13,28)(14,29)(15,30)(16,32)(18,35)(19,26)(20,36)(21,34)(22,37)(24,27)(25,39)(31,40)(33,38)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325533);