#482101

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#36020	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

1081 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)(4,7)(6,8)(9,10)(11,19)(12,23,25,16)(13,17)(14,18)(15,20,29,36)(21,33)(22,34)(24,31,26,30)(27,32)(28,35,40,37)(38,39);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,18,29)(12,20,25,36)(13,21,27,38)(16,19,23,34)(17,28,32,40)(24,35,26,37)(30,33,31,39);(9,12)(10,16)(11,20)(13,26)(14,25)(15,19)(17,31)(18,23)(21,37)(22,36)(24,27)(28,39)(29,34)(30,32)(33,40)(35,38);(9,13,14,27)(10,17,18,32)(11,21,22,38)(12,24,25,26)(15,28,29,40)(16,30,23,31)(19,33,34,39)(20,35,36,37)

Polycyclic presentation:

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

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, 389297);