#481822

order	512
nilpotency class	4
isoclinism	isoclinic class of #481819
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1065 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)(12,16)(13,26,27,17)(14,18)(15,21,29,36)(20,33)(22,34)(23,37)(24,39,40,31)(25,32)(28,35,30,38);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,18,29)(12,23,25,20)(13,21,27,36)(16,30,32,28)(17,19,26,34)(24,38,40,35)(31,37,39,33);(2,6)(5,7);(9,12,14,25)(10,16,18,32)(11,20,22,23)(13,24,27,40)(15,28,29,30)(17,31,26,39)(19,33,34,37)(21,35,36,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,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, 389018);