#8818371

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

4245 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725567);