#9007818

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=g,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, 8915014);