#10055756

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	#47093	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9962952);