#9243504

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	#53704	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9150700);