#9007844

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	#51133	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,e^2=hi,f^2=h,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, 8915040);