#480856

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

825 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)(19,27,30,42)(20,24)(21,34,36,25)(22,26)(23,29)(28,43,31,45)(32,40)(33,48,47,41)(35,38)(37,44)(39,46);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,19,22,30)(18,23,26,38)(20,31,32,28)(21,35,36,29)(24,39,40,37)(25,42,34,27)(33,44,47,46)(41,43,48,45);(3,10)(7,14)(11,12)(15,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,41)(26,40)(27,43)(29,44)(30,31)(34,48)(35,46)(36,47)(38,39)(42,45);(1,4,6,5)(2,8,9,13)(3,11,10,12)(7,15,14,16)(20,32)(24,40)(28,31)(33,47)(37,39)(41,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,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, 388052);