#8837070

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	#42266	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8744266);