#8339949

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

4133 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8247145);