#480701

order	512
nilpotency class	4
isoclinism	isoclinic class of #480670
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, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

816 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,d^b=di,f^b=fg,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, 387897);