#8147848

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

4245 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=i,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, 8055044);