#9059389

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^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, 8966585);