#9059122

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	#50711	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=h,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, 8966318);