#9040612

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

4118 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947808);