#10047015

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

4001 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)(4,9)(5,10)(6,8)(11,14)(12,15)(13,16)(17,18,23,24)(19,26,22,21)(20,25,29,31)(27,32,28,30)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,3,6,10)(2,7,8,5)(4,12,13,11)(9,15,16,14)(17,19,23,22)(18,21,24,26)(20,28,29,27)(25,32,31,30)(35,36)(39,42)(43,44)(47,48);(2,8)(3,10)(4,13)(14,15)(18,24)(20,29)(21,26)(27,28)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3)(2,5)(4,11)(6,10)(7,8)(9,15)(12,13)(14,16)(17,20)(18,25)(19,27)(21,30)(22,28)(23,29)(24,31)(26,32)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4,6,13)(2,9,8,16)(3,11,10,12)(5,15,7,14)(17,21)(18,19)(20,30)(22,24)(23,26)(25,27)(28,31)(29,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,5)(2,3)(4,14)(6,7)(8,10)(9,12)(11,16)(13,15)(17,22,23,19)(18,26,24,21)(20,28,29,27)(25,32,31,30)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=ghi,e^2=gh,f^2=g,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, 9954211);