#361137

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1282 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)(3,7)(4,5)(6,8)(9,10)(11,19,23,15)(12,24,26,16)(13,18)(14,17)(20,30)(21,31)(22,35,27,32)(25,36,28,33)(29,34)(37,39)(38,40);(2,5)(6,7)(9,11)(10,15)(12,21)(13,27)(14,23)(16,31)(17,19)(18,35)(20,25)(22,29)(24,38)(26,40)(28,39)(30,33)(32,34)(36,37);(9,12)(10,16)(11,20)(13,28)(14,26)(15,30)(17,24)(18,36)(19,37)(21,27)(22,40)(23,39)(25,29)(31,35)(32,38)(33,34);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,21)(13,25)(14,26)(15,31)(17,24)(18,33)(19,38)(20,27)(22,39)(23,40)(28,29)(30,35)(32,37)(34,36)

Polycyclic presentation:

2^9|a^2=h,d^2=ef,g^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, 268333);