#361139

order	512
nilpotency class	3
isoclinism	isoclinic class of #222256
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1320 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)(3,7)(4,5)(6,8)(9,10,14,20,15,17,13,19)(11,21,22,30,25,16,24,32)(12,26,27,34,28,18,29,33)(23,31,37,39,38,35,36,40);(2,5)(6,7)(9,11)(10,16)(12,23)(13,22)(14,24)(15,25)(17,21)(18,31)(19,32)(20,30)(26,35)(27,36)(28,38)(29,37)(33,40)(34,39);(10,17)(11,22)(13,14)(16,30)(18,26)(21,32)(23,36)(24,25)(27,29)(31,39)(35,40)(37,38);(1,3,4,8)(2,6,5,7)(9,12)(10,18)(11,23)(13,27)(14,29)(15,28)(16,31)(17,26)(19,33)(20,34)(21,35)(22,36)(24,37)(25,38)(30,39)(32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268335);