#8943896

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	#43651	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(3,6)(4,14)(5,10)(7,9)(8,11)(12,15)(13,16)(17,18,22,25)(19,23,29,32)(20,24,30,21)(26,27,28,31)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,3,7,11)(2,8,9,6)(4,12,10,16)(5,15,14,13)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,32)(26,30)(37,45)(40,44)(42,48)(46,47);(2,9)(5,14)(6,8)(13,15)(17,20,22,30)(18,24,25,21)(19,26,29,28)(23,27,32,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,4)(2,5)(3,12)(6,13)(7,10)(8,15)(9,14)(11,16)(17,21)(18,20)(19,27)(22,24)(23,28)(25,30)(26,32)(29,31)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,5)(2,10)(3,13)(4,9)(6,16)(7,14)(8,12)(11,15)(17,20)(18,24)(19,28)(21,25)(22,30)(23,31)(26,29)(27,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48);(1,6,7,8)(2,11,9,3)(4,13,10,15)(5,16,14,12)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,32)(26,30)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851092);