#482001

order	512
nilpotency class	4
isoclinism	isoclinic class of #481837
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

906 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18)(19,27)(20,24)(21,33,34,25)(22,26)(23,29,36,43)(28,39)(30,40)(31,37)(32,45,44,38)(35,42,46,48)(41,47);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,30)(18,23,26,36)(20,28,31,41)(21,29,34,43)(24,35,37,46)(25,27,33,40)(32,42,44,48)(38,39,45,47);(2,8)(7,15)(9,10)(11,16)(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,38)(26,37)(27,39)(29,42)(30,41)(33,45)(34,44)(36,46)(40,47)(43,48);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(20,31)(24,37)(28,41)(32,44)(35,46)(38,45)(39,47)(42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,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, 389197);