#480564

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
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

859 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:

(1,2,5,9,6,10,4,8)(3,7,11,14,12,15,13,16)(17,18)(19,27,30,40)(20,24)(21,33,34,25)(22,26)(23,29)(28,39,41,47)(31,37)(32,45,44,38)(35,42)(36,43)(46,48);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,14)(10,15)(17,19)(18,23,26,36)(20,28)(21,29)(22,30)(24,35,37,46)(25,27,33,40)(31,41)(32,42)(34,43)(38,39,45,47)(44,48);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,15,16)(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);(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=g,c^2=h,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ch,f^b=fgi,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, 387760);