#9030148

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	#53679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3906 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)(12,13)(14,16)(17,18,23,26)(19,24,22,20)(21,25,31,32)(27,29,30,28)(33,34)(35,36)(37,45)(38,39)(40,41)(42,46)(43,47)(44,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19,23,22)(18,24,26,20)(21,27,31,30)(25,29,32,28)(34,39)(35,41)(42,45)(43,48);(2,6)(3,5)(4,12)(15,16)(17,20,23,24)(18,19,26,22)(21,29,31,28)(25,30,32,27)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,20)(18,19)(21,29)(22,26)(23,24)(25,30)(27,32)(28,31)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,22,23,19)(18,20,26,24)(21,30,31,27)(25,28,32,29)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=h,e^2=gh,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, 8937344);