#9206528

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	#57437	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113724);