#8933767

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	#47231	C₂
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) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8840963);