#9017174

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	#51769	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(17,18)(19,24)(20,25)(21,30)(22,23)(26,32)(27,31)(28,29)(34,36)(35,42)(38,46)(45,47);(2,7)(4,14)(8,11)(10,15)(17,18)(19,24)(20,28)(21,26)(22,23)(25,29)(27,31)(30,32)(33,34,39,36)(35,37,42,40)(38,41,46,44)(43,45,48,47);(2,7)(3,12)(4,14)(5,16)(18,23)(19,27)(20,29)(21,32)(33,35,39,42)(34,40,36,37)(38,43,46,48)(41,47,44,45);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,19,22,27)(18,24,23,31)(20,30,29,26)(21,28,32,25)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,20)(18,25)(19,26)(21,31)(22,29)(23,28)(24,32)(27,30)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,21,22,32)(18,26,23,30)(19,28,27,25)(20,31,29,24)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=i,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8924370);