#8926639

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3983 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=g,e^2=g,f^2=h,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, 8833835);