#8734666

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

4118 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18)(19,21)(20,25)(22,26)(23,27)(24,30)(28,31)(29,32)(33,34)(35,43)(36,47)(37,41)(38,42)(39,46)(40,45)(44,48);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,26)(31,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(3,11)(4,12)(7,16)(8,13)(17,20,23,29)(18,25,27,32)(19,28,30,22)(21,31,24,26)(33,34,38,42)(35,39,46,43)(36,40,45,47)(37,41,44,48);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,20)(18,25)(19,28)(21,31)(22,30)(23,29)(24,26)(27,32)(33,34)(35,43)(36,40)(37,48)(38,42)(39,46)(41,44)(45,47);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,21,23,24)(18,19,27,30)(20,31,29,26)(22,25,28,32)(33,36)(34,40)(35,44)(37,46)(38,45)(39,48)(41,43)(42,47);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,22,23,28)(18,26,27,31)(19,29,30,20)(21,32,24,25)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=gh,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, 8641862);