#387736

order	512
nilpotency class	3
isoclinism	isoclinic class of #213818
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

1084 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)(4,6)(7,8)(9,10)(11,19,22,15)(12,23,25,38)(13,17)(14,18)(16,24,31,40)(20,34)(21,32,26,29)(27,33)(28,35)(30,39)(36,37);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,20)(13,26)(14,22)(16,30)(17,32)(18,19)(21,27)(23,37)(24,39)(25,36)(28,31)(29,33)(34,38)(35,40);(1,4,5,8)(2,6,3,7)(9,12,14,25)(10,16)(11,20,22,36)(13,24,27,40)(15,28)(17,23)(18,31)(19,30)(21,35,26,39)(29,37)(32,34)(33,38);(2,3)(6,7)(10,17,18,33)(15,29,19,32)(16,23,31,38)(28,37,30,34)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=ei,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=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, 294932);