#8734327

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

4221 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(33,34)(35,44)(36,42)(37,45)(38,40)(39,41)(43,47)(46,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,32)(29,31)(33,35,39,38)(34,40,41,44)(36,45,46,43)(37,42,47,48);(2,3)(7,9)(8,10)(15,16)(20,28)(23,31)(27,32)(29,30)(34,41)(36,46)(40,44)(43,45);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,22,28)(18,23,26,31)(19,27,25,32)(21,30,24,29)(33,36)(34,42)(35,43)(37,40)(38,45)(39,46)(41,48)(44,47);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,19)(18,24)(20,27)(21,26)(22,25)(23,29)(28,32)(30,31)(33,37)(34,43)(35,42)(36,40)(38,48)(39,47)(41,45)(44,46);(1,5)(2,8)(3,10)(4,11)(6,14)(7,15)(9,16)(12,13)(17,21,22,24)(18,25,26,19)(20,29,28,30)(23,27,31,32)(33,38,39,35)(34,40,41,44)(36,43,46,45)(37,42,47,48)

Polycyclic presentation:

2^9|b^2=hi,d^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, 8641523);