#8359734

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

4175 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8266930);