#8963341

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

4113 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870537);