#8534631

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

4306 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:

(1,2,6,7)(3,11,12,8)(4,14,15,9)(5,10,16,13)(17,18,23,25)(19,24,21,20)(22,26,31,32)(27,30,29,28)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48);(2,7)(3,12)(9,14)(10,13)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,30,32,28)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(2,7)(4,15)(8,11)(10,13)(18,25)(19,21)(26,32)(27,29)(33,35,38,36)(34,40,42,39)(37,43,45,44)(41,47,48,46);(1,3)(2,8)(4,13)(5,9)(6,12)(7,11)(10,15)(14,16)(17,20,23,24)(18,19,25,21)(22,30,31,28)(26,29,32,27)(33,36,38,35)(34,40,42,39)(37,43,45,44)(41,46,48,47);(1,4)(2,9)(3,13)(5,8)(6,15)(7,14)(10,12)(11,16)(17,21,23,19)(18,24,25,20)(22,29,31,27)(26,30,32,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5)(2,10)(3,14)(4,8)(6,16)(7,13)(9,12)(11,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(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=gh,c^2=ghi,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8441827);