#8741948

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

4113 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^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=dh,d^c=dgi,f^d=fgi,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, 8649144);