#8366595

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

4074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=h,e^2=ghi,f^2=gh,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, 8273791);