#8930501

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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)(10,15)(11,19)(12,22)(18,23)(25,26)(27,33)(28,34)(29,35)(30,36)(31,32)(37,40)(38,49)(39,43)(41,50)(42,47)(44,45)(46,48)(51,53)(52,55)(54,56);(1,3,4,7)(2,5,6,8)(9,10,14,15)(11,20,19,16)(12,17,22,24)(13,23,21,18)(26,32)(29,33)(34,47)(36,48)(40,43)(44,49)(50,56)(53,55);(1,2)(3,5)(4,6)(7,8)(9,11,14,19)(10,16,15,20)(12,21,22,13)(17,23,24,18)(25,27,31,35)(26,33,32,29)(28,39,42,37)(30,45,46,38)(34,43,47,40)(36,44,48,49)(41,51,54,52)(50,53,56,55);(9,12)(10,17)(11,21)(13,19)(14,22)(15,24)(16,23)(18,20)(25,28,31,42)(26,34,32,47)(27,37,35,39)(29,43,33,40)(30,41,46,54)(36,50,48,56)(38,51,45,52)(44,55,49,53);(1,3)(2,5)(4,7)(6,8)(9,13,14,21)(10,18,15,23)(11,12,19,22)(16,17,20,24)(25,29,31,33)(26,35,32,27)(28,40,42,43)(30,44,46,49)(34,37,47,39)(36,45,48,38)(41,53,54,55)(50,51,56,52);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,21)(13,19)(14,22)(15,24)(16,23)(18,20)(25,30)(26,36)(27,38)(28,41)(29,44)(31,46)(32,48)(33,49)(34,50)(35,45)(37,51)(39,52)(40,53)(42,54)(43,55)(47,56)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8837697);