#480575

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

848 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=hi,d^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ch,e^b=ei,f^b=fg,d^c=di,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387771);