#9059032

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
prime radical	#54807	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8966228);