#391805

order	512
nilpotency class	3
isoclinism	isoclinic class of #213798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
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

1215 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)(9,10,15,19)(11,16,22,34)(12,17,26,13)(14,29,31,18)(20,37,39,32)(21,33,27,24)(23,38,40,35)(25,28,30,36);(2,5)(3,7)(9,11)(10,16)(12,24,26,33)(13,27,17,21)(14,23)(15,22)(18,35)(19,34)(20,25,39,30)(28,32,36,37)(29,38)(31,40);(3,7)(6,8)(9,12)(10,17)(11,20)(13,19)(14,30)(15,26)(16,32)(18,28)(21,38)(22,39)(23,33)(24,40)(25,31)(27,35)(29,36)(34,37);(1,3,4,7)(2,6,5,8)(9,13,15,17)(10,12,19,26)(11,21,22,27)(14,28,31,36)(16,33,34,24)(18,25,29,30)(20,38,39,35)(23,37,40,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299001);