#9243514

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	#56650	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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)(3,9)(4,14)(5,10)(6,11)(7,8)(12,13)(15,16)(17,18,22,25)(19,20,23,28)(21,24,30,32)(26,27,29,31)(33,34)(35,42)(36,46)(37,41)(38,39)(40,45)(43,44)(47,48);(2,8)(6,10)(9,14)(13,16)(17,19,22,23)(18,20,25,28)(21,29,30,26)(24,31,32,27)(34,38)(37,46)(40,42)(44,48);(1,3,7,4)(2,9,8,14)(5,15,11,12)(6,13,10,16)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32)(33,35)(34,40)(36,47)(37,48)(38,42)(39,45)(41,43)(44,46);(1,4,7,3)(2,9,8,14)(5,15,11,12)(6,16,10,13)(33,36)(34,37)(35,43)(38,46)(39,41)(40,44)(42,48)(45,47);(1,5,7,11)(2,10,8,6)(3,12,4,15)(9,13,14,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,29)(26,32)(30,31)(33,37,39,46)(34,41,38,36)(35,44,45,48)(40,47,42,43);(1,6,7,10)(2,11,8,5)(3,13,4,16)(9,12,14,15)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,32)(28,31)(33,38,39,34)(35,42,45,40)(36,46,41,37)(43,48,47,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 9150710);