#8943996

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

4075 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)(3,9)(4,13)(5,7)(6,10)(8,11)(12,15)(14,16)(17,18,22,25)(19,23,29,32)(20,24,30,21)(26,27,28,31)(33,34,39,37)(35,42,40,36)(38,45,47,41)(43,44,48,46);(1,3,8,12)(2,9,11,15)(4,14,5,10)(6,13,16,7)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,32)(26,30)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,4,8,5)(2,7,11,13)(3,10,12,14)(6,9,16,15)(17,20,22,30)(18,24,25,21)(19,26,29,28)(23,27,32,31)(33,36,39,42)(34,40,37,35)(38,46,47,44)(41,43,45,48);(1,5,8,4)(2,7,11,13)(3,10,12,14)(6,15,16,9)(17,21)(18,20)(19,27)(22,24)(23,28)(25,30)(26,32)(29,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,6)(2,10)(3,7)(4,15)(5,9)(8,16)(11,14)(12,13)(17,20)(18,24)(19,28)(21,25)(22,30)(23,31)(26,29)(27,32)(33,37,39,34)(35,36,40,42)(38,45,47,41)(43,44,48,46);(1,7)(2,4)(3,6)(5,11)(8,13)(9,14)(10,15)(12,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,32)(26,30)(33,38,39,47)(34,41,37,45)(35,43,40,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,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, 8851192);