#482400

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	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1035 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)(10,16)(12,22)(13,23)(14,29)(15,24)(17,30)(18,21)(19,34)(20,31)(25,40)(26,37)(27,39)(28,38)(32,36)(33,35);(2,6)(5,7)(9,12,15,27)(10,17,20,25)(11,22,24,39)(13,28,14,26)(16,30,31,40)(18,33,19,32)(21,35,34,36)(23,38,29,37);(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,c^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, 389596);