#9059124

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(17,18,22,24)(19,23,20,25)(21,26,29,31)(27,30,28,32)(33,34,38,40)(35,41,42,36)(37,39,45,46)(43,47,48,44);(2,7)(3,12)(9,13)(10,15)(17,19)(18,23)(20,22)(21,27)(24,25)(26,30)(28,29)(31,32)(33,34)(35,41)(36,42)(37,39)(38,40)(43,47)(44,48)(45,46);(2,7)(3,12)(4,14)(5,16)(18,24)(19,20)(21,29)(30,32)(35,42)(36,41)(37,45)(39,46);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,20,22,19)(18,25,24,23)(21,27,29,28)(26,30,31,32)(33,35,38,42)(34,36,40,41)(37,43,45,48)(39,44,46,47);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,21,22,29)(18,26,24,31)(19,27,20,28)(23,30,25,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,47)(46,48);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,32)(29,31)(33,37)(34,39)(35,43)(36,44)(38,45)(40,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,d^2=gh,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, 8966320);