#8228325

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

4216 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,18)(19,27)(20,29)(21,26)(22,24)(23,30)(25,31)(28,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48);(2,7)(4,8)(9,14)(12,15)(17,19,22,30)(18,23,24,27)(20,26,31,28)(21,29,32,25)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(5,13)(9,14)(10,16)(12,15)(18,24)(21,32)(23,27)(25,29)(34,38)(39,41)(40,44)(46,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,20,22,31)(18,25,24,29)(19,28,30,26)(21,23,32,27)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,21,22,32)(18,26,24,28)(19,29,30,25)(20,23,31,27)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,e^2=ghi,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

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, 8135521);