#8418657

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

3997 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325853);