#8147859

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8055055);