#9007416

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

3835 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)(19,27)(20,25)(21,26)(22,23)(24,28)(29,32)(30,31)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(1,3,7,5)(2,8,10,6)(4,11,12,14)(9,15,16,13)(19,28)(21,29)(24,27)(26,32)(34,39)(35,42)(41,44)(43,48);(3,5)(4,12)(6,8)(9,16)(17,19,23,28)(18,24,22,27)(20,21,31,29)(25,32,30,26)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,20,23,31)(18,25,22,30)(19,29,28,21)(24,26,27,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,5,7,3)(2,6,10,8)(4,11,12,14)(9,15,16,13)(17,21,23,29)(18,26,22,32)(19,20,28,31)(24,30,27,25)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,6)(2,3)(4,13)(5,10)(7,8)(9,11)(12,15)(14,16)(17,22,23,18)(19,24,28,27)(20,30,31,25)(21,32,29,26)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gh,e^2=gi,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, 8914612);