#482402

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
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	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1088 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)(11,21,24,34)(12,25)(13,19,14,18)(15,20)(16,23)(17,27)(22,36,39,35)(26,32,28,33)(29,31)(30,37)(38,40);(2,5)(3,8)(6,7)(9,11,15,24)(10,16,20,31)(12,22,27,39)(13,23,14,29)(17,30,25,40)(18,21,19,34)(26,37,28,38)(32,36,33,35);(2,6)(5,7)(9,12)(10,17)(11,22)(13,28)(14,26)(15,27)(16,30)(18,33)(19,32)(20,25)(21,35)(23,38)(24,39)(29,37)(31,40)(34,36);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,23,24,29)(12,26,27,28)(16,21,31,34)(17,32,25,33)(22,37,39,38)(30,36,40,35)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,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, 389598);