#9236455

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3818 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)(3,10,8,4)(6,13,14,9)(11,12,15,16)(18,23)(21,24)(25,30)(29,31)(33,34)(35,41)(36,40)(37,42)(38,43)(39,44)(45,46)(47,48);(2,5)(3,8)(9,13)(11,15)(20,28)(25,30)(27,32)(29,31)(33,35)(34,37)(36,45)(38,46)(39,42)(40,47)(41,44)(43,48);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,16)(14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,36,39,43)(34,40,44,38)(35,45,42,48)(37,47,41,46);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29)(33,35)(34,41)(36,45)(37,44)(38,47)(39,42)(40,46)(43,48);(1,5,7,2)(3,10,8,4)(6,13,14,9)(11,16,15,12)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,37,39,41)(34,42,44,35)(36,47,43,46)(38,45,40,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)(33,38,39,40)(34,43,44,36)(35,46,42,47)(37,48,41,45)

Polycyclic presentation:

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

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, 9143651);