#10506276

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

4034 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:

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

Polycyclic presentation:

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

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, 10413472);