#9243517

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	#2428	C₂²
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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=gh,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, 9150713);