#8270567

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

4076 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=i,e^2=hi,f^2=i,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, 8177763);