#8756996

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

4121 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664192);