#8269730

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

4216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 8176926);