#8159605

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

4144 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066801);