#9029572

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	#53934	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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)(5,14)(6,7)(10,16)(11,15)(12,13)(17,18,23,26)(19,24,22,20)(21,25,31,32)(27,29,30,28)(33,34)(35,44)(36,37)(38,46)(39,40)(41,45)(42,48)(43,47);(1,2)(3,8)(4,12)(5,10)(6,7)(9,13)(11,15)(14,16)(17,19,23,22)(18,24,26,20)(21,27,31,30)(25,29,32,28)(34,40)(35,45)(37,42)(43,46);(2,7)(3,11)(4,13)(5,16)(17,20,23,24)(18,19,26,22)(21,29,31,28)(25,30,32,27)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,20)(18,19)(21,29)(22,26)(23,24)(25,30)(27,32)(28,31)(33,36)(34,42)(35,46)(37,40)(38,41)(39,48)(43,45)(44,47);(1,4)(2,9)(3,10)(5,15)(6,13)(7,12)(8,16)(11,14)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(33,37,39,42)(34,36,40,48)(35,47,45,38)(41,46,44,43);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,22,23,19)(18,20,26,24)(21,30,31,27)(25,28,32,29)(33,38,39,47)(34,43,40,46)(35,42,45,37)(36,41,48,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=gh,e^2=g,f^2=i,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, 8936768);