#8734162

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

4335 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)(3,8)(4,7)(5,9)(6,10)(11,13)(12,15)(14,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,37,40)(35,41,42,38)(36,39,46,45)(43,48,44,47);(1,3)(2,5)(4,11)(6,9)(7,14)(8,10)(12,16)(13,15)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(35,42)(38,41)(43,44)(47,48);(4,12)(7,15)(11,16)(13,14)(19,27)(20,28)(23,32)(24,29)(33,35,37,42)(34,38,40,41)(36,43,46,44)(39,47,45,48);(1,4,6,12)(2,7,10,15)(3,11,9,16)(5,14,8,13)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,35)(34,38)(36,44)(37,42)(39,48)(40,41)(43,46)(45,47);(1,3)(2,8)(4,11)(5,10)(6,9)(7,13)(12,16)(14,15)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,34)(35,38)(36,45)(37,40)(39,46)(41,42)(43,48)(44,47);(1,5,6,8)(2,9,10,3)(4,13,12,14)(7,11,15,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,36)(34,39)(35,43)(37,46)(38,47)(40,45)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=g,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, 8641358);