#480669

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27)(20,24)(21,34,35,25)(22,26)(23,30,36,43)(28,41)(29,42)(31,45)(32,47,48,39)(33,40)(37,44,38,46);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19,22,28)(18,23,26,36)(20,31,33,29)(21,30,35,43)(24,38,40,37)(25,27,34,41)(32,46,48,44)(39,45,47,42);(3,9)(6,12)(10,15)(13,16)(19,28)(23,36)(27,41)(29,31)(30,43)(37,38)(42,45)(44,46);(2,7)(6,12)(8,14)(13,16)(17,20,22,33)(18,24,26,40)(19,29,28,31)(21,32,35,48)(23,37,36,38)(25,39,34,47)(27,42,41,45)(30,44,43,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,d^b=di,f^b=fg,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, 387865);