#482652

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

902 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,f^d=fi,f^e=fi

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, 389848);