#8993007

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

3846 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^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, 8900203);