#9040088

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	#56369	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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)(9,10,15,13)(11,19,20,16)(12,18,14,17)(21,23,22,24)(26,31)(27,35)(29,42)(33,38)(36,39)(40,54)(46,51)(55,56);(3,7)(5,8)(10,13)(11,20)(17,18)(21,22)(26,31)(32,44)(33,38)(34,43)(45,47)(46,51)(48,52)(55,56);(1,2)(3,5)(4,6)(7,8)(9,11,15,20)(10,16,13,19)(12,21,14,22)(17,23,18,24)(25,26)(27,32)(28,38)(29,43)(30,31)(33,41)(34,42)(35,44)(36,47)(37,51)(39,45)(40,48)(46,49)(50,55)(52,54)(53,56);(9,12,15,14)(10,17,13,18)(11,21,20,22)(16,23,19,24)(25,27,30,35)(26,32,31,44)(28,39,41,36)(29,37,42,49)(33,47,38,45)(34,46,43,51)(40,53,54,50)(48,56,52,55);(1,3)(2,5)(4,7)(6,8)(9,13,15,10)(11,19,20,16)(12,18,14,17)(21,24,22,23)(25,28,30,41)(26,33,31,38)(27,36,35,39)(29,40,42,54)(32,45,44,47)(34,48,43,52)(37,50,49,53)(46,55,51,56);(1,2)(3,5)(4,6)(7,8)(9,14,15,12)(10,18,13,17)(11,22,20,21)(16,24,19,23)(25,29)(26,34)(27,37)(28,40)(30,42)(31,43)(32,46)(33,48)(35,49)(36,50)(38,52)(39,53)(41,54)(44,51)(45,55)(47,56)

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=i,e^2=i,f^2=g,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, 8947284);