#8407189

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

4125 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,24,29,31)(20,21,26,25)(27,32,30,28)(33,34)(35,40)(36,46)(37,41)(38,42)(39,43)(44,48)(45,47);(2,6)(3,9)(8,13)(11,15)(18,23)(20,26)(24,31)(27,30)(33,35,39,44)(34,40,43,48)(36,47,41,38)(37,42,46,45);(4,12)(8,13)(11,15)(14,16)(18,23)(21,25)(24,31)(28,32)(35,44)(38,47)(40,48)(42,45);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,19,22,29)(18,24,23,31)(20,30,26,27)(21,28,25,32)(33,36)(34,37)(35,38)(39,41)(40,45)(42,48)(43,46)(44,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,22,26)(18,25,23,21)(19,27,29,30)(24,28,31,32)(33,37)(34,41)(35,45)(36,43)(38,48)(39,46)(40,47)(42,44);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,25)(18,26,23,20)(19,28,29,32)(24,27,31,30)(33,38)(34,42)(35,36)(37,48)(39,47)(40,46)(41,44)(43,45)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=g,e^2=h,f^2=g,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, 8314385);