#8898527

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

4288 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,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,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, 8805723);