#9030008

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	#42065	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) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8937204);