#9040095

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8947291);