#9054391

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	#42153	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=g,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, 8961587);