#8417311

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

3973 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324507);