#8811209

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

4068 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8718405);