#8963469

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=hi,e^2=ghi,f^2=h,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, 8870665);