#480847

order	512
nilpotency class	4
isoclinism	isoclinic class of #480690
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×C₄⋊C₄	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,e^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,e^b=ei,f^b=fgi,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, 388043);