#383702

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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

1390 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)(3,5)(4,7)(6,8)(9,10)(11,19,22,15)(12,23,25,16)(13,18)(14,17)(20,32,27,28)(21,29)(24,36,37,30)(26,31)(33,35)(34,39)(38,40);(1,3)(2,4)(5,8)(6,7)(11,20)(13,26)(15,28)(18,31)(19,32)(21,34)(22,27)(24,37)(29,39)(30,36)(33,40)(35,38);(1,4,6,5)(2,7,8,3)(9,11)(10,15)(12,21)(13,27)(14,22)(16,29)(17,19)(18,32)(20,26)(23,33)(24,38)(25,35)(28,31)(30,40)(34,37)(36,39);(1,5,6,4)(2,7,8,3)(9,12,14,25)(10,16,17,23)(11,21,22,35)(13,24,26,37)(15,29,19,33)(18,30,31,36)(20,34,27,38)(28,39,32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290898);