#9053942

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	#51170	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961138);