#8926782

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	C₂×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833978);