#8926620

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	#56739	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(4,8)(6,7)(11,17)(15,18)(19,23)(22,24)(25,26,31,29)(27,35,36,32)(28,39,41,33)(30,42,44,34)(37,45,50,49)(38,46,43,48)(40,47,54,52)(51,56,53,55);(2,7)(3,8)(9,10,14,12)(11,15,17,18)(13,20,21,16)(19,24,23,22)(27,36)(32,35)(37,50)(38,43)(45,49)(46,48)(51,53)(55,56);(1,3,6,8)(2,4,7,5)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,32)(28,37)(29,35)(30,43)(31,36)(33,45)(34,48)(38,44)(39,49)(40,53)(41,50)(42,46)(47,56)(51,54)(52,55);(1,4,6,5)(2,3,7,8)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,28,31,41)(26,33,29,39)(27,37,36,50)(30,40,44,54)(32,45,35,49)(34,47,42,52)(38,51,43,53)(46,55,48,56);(1,2)(3,4)(5,8)(6,7)(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,29,31,26)(27,35,36,32)(28,33,41,39)(30,42,44,34)(37,45,50,49)(38,48,43,46)(40,47,54,52)(51,55,53,56);(1,5,6,4)(2,8,7,3)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,30)(26,34)(27,38)(28,40)(29,42)(31,44)(32,46)(33,47)(35,48)(36,43)(37,51)(39,52)(41,54)(45,55)(49,56)(50,53)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=h,e^2=gh,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, 8833816);