#10055854

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

4322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9963050);