#8926531

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	#55815	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(3,5)(4,7)(6,8)(10,14)(12,19)(15,20)(18,23)(25,26,31,32)(27,33,39,46)(28,34,42,47)(29,35,43,30)(36,48,53,56)(37,38,49,45)(40,41,44,50)(51,55,54,52);(1,2,6,8)(3,5,7,4)(11,17)(15,20)(18,23)(22,24)(26,32)(29,43)(33,46)(34,47)(37,49)(40,44)(48,56)(51,54);(1,3,6,7)(2,4,8,5)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27)(26,33)(28,36)(29,37)(30,45)(31,39)(32,46)(34,48)(35,38)(40,51)(41,55)(42,53)(43,49)(44,54)(47,56)(50,52);(1,4,6,5)(2,3,8,7)(25,28,31,42)(26,34,32,47)(27,36,39,53)(29,44,43,40)(30,41,35,50)(33,48,46,56)(37,54,49,51)(38,52,45,55);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,29)(26,35)(27,37)(28,40)(30,32)(31,43)(33,38)(34,41)(36,51)(39,49)(42,44)(45,46)(47,50)(48,55)(52,56)(53,54);(1,5,6,4)(2,7,8,3)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,35)(26,29,32,43)(27,38,39,45)(28,41,42,50)(33,49,46,37)(34,44,47,40)(36,52,53,55)(48,51,56,54)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=g,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, 8833727);