#9054441

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	#52872	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 56 points:

(1,2,5,3)(4,7,8,6)(10,13)(15,18)(16,21)(22,24)(27,35)(31,36)(37,40)(38,49)(45,47)(46,50)(51,53)(55,56);(2,3)(6,7)(12,20)(16,21)(17,23)(22,24)(25,26,30,34)(27,31,35,36)(28,39,42,32)(29,33,44,43)(37,47,40,45)(38,46,49,50)(41,52,54,48)(51,56,53,55);(1,3,5,2)(4,7,8,6)(9,10,14,13)(11,15,19,18)(12,16,20,21)(17,22,23,24)(25,26)(27,36)(28,39)(29,43)(30,34)(31,35)(32,42)(33,44)(37,45)(38,46)(40,47)(41,48)(49,50)(51,56)(52,54)(53,55);(1,4,5,8)(2,6,3,7)(9,11,14,19)(10,15,13,18)(12,17,20,23)(16,22,21,24)(25,27)(26,31)(28,40)(29,38)(30,35)(32,47)(33,46)(34,36)(37,42)(39,45)(41,53)(43,50)(44,49)(48,56)(51,54)(52,55);(9,12,14,20)(10,16,13,21)(11,17,19,23)(15,22,18,24)(25,28,30,42)(26,32,34,39)(27,37,35,40)(29,41,44,54)(31,45,36,47)(33,48,43,52)(38,51,49,53)(46,55,50,56);(9,13,14,10)(11,18,19,15)(12,21,20,16)(17,24,23,22)(25,29)(26,33)(27,38)(28,41)(30,44)(31,46)(32,48)(34,43)(35,49)(36,50)(37,51)(39,52)(40,53)(42,54)(45,55)(47,56)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8961637);