#372421

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1304 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)(3,6)(4,5)(7,8)(9,10,13,18,14,19,30,17)(11,20,22,32,23,35,29,15)(12,24,27,34,28,16,26,33)(21,36,38,31,25,40,37,39);(9,11)(10,15)(12,25,28,21)(13,29)(14,23)(16,31,24,39)(17,20)(18,35)(19,32)(22,30)(26,37,27,38)(33,36,34,40);(2,5)(6,8)(9,12)(10,16)(11,21)(13,26)(14,28)(15,31)(17,33)(18,34)(19,24)(20,36)(22,37)(23,25)(27,30)(29,38)(32,39)(35,40);(1,3,4,7)(2,6,5,8)(10,17)(13,30)(15,20)(16,33)(18,19)(22,29)(24,34)(26,27)(31,36)(32,35)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,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, 279617);