#215574

order	512
nilpotency class	3
isoclinism	isoclinic class of #162948
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122770);