#9057309

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

4304 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)(4,11)(5,13)(6,8)(9,14)(10,16)(12,15)(19,25)(23,26)(27,31)(30,32)(33,34)(35,37)(36,41)(38,40)(39,42)(43,46)(44,45)(47,48);(1,3,6,4)(2,7,8,11)(5,10,14,12)(9,15,13,16)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35,38,42)(34,39,40,37)(36,43,44,48)(41,47,45,46);(2,8)(3,4)(5,14)(15,16)(18,22)(19,25)(20,28)(30,32)(34,40)(36,44)(37,39)(43,48);(1,4,6,3)(2,7,8,11)(5,12,14,10)(9,15,13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,34)(35,39)(36,45)(37,42)(38,40)(41,44)(43,46)(47,48);(1,2)(3,7)(4,11)(5,9)(6,8)(10,15)(12,16)(13,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,38,44)(34,41,40,45)(35,43,42,48)(37,46,39,47);(1,5,6,14)(2,9,8,13)(3,10,4,12)(7,15,11,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,39)(34,42,40,35)(36,46,44,47)(41,48,45,43)

Polycyclic presentation:

2^9|b^2=gi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8964505);