#372420

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1226 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)(9,10,14,18,15,20,13,19)(11,21,25,16,24,32,23,33)(12,26,28,34,29,17,30,35)(22,36,37,39,27,40,38,31);(9,11)(10,16)(12,27,29,22)(13,25)(14,23)(15,24)(17,31,26,39)(18,21)(19,32)(20,33)(28,37,30,38)(34,36,35,40);(2,5)(6,8)(9,12)(10,17)(11,22)(13,28)(14,30)(15,29)(16,31)(18,34)(19,35)(20,26)(21,36)(23,37)(24,27)(25,38)(32,40)(33,39);(1,3,4,7)(2,6,5,8)(10,18)(13,14)(16,21)(17,34)(19,20)(23,25)(26,35)(28,30)(31,36)(32,33)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=fg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi,e^d=ei,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, 279616);