#8926766

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

4038 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:

(2,5)(6,8)(9,10,14,18)(11,17,21,15)(12,22,16,13)(19,24,23,20)(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);(3,7)(6,8)(9,11,14,21)(10,15,18,17)(12,23,16,19)(13,20,22,24)(27,36)(32,35)(37,50)(38,43)(45,49)(46,48)(51,53)(55,56);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,13,18,22)(11,19,21,23)(15,24,17,20)(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);(9,13)(10,16)(11,20)(12,18)(14,22)(15,19)(17,23)(21,24)(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);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,19)(13,20)(14,21)(15,18)(16,23)(22,24)(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);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,20)(12,18)(14,22)(15,19)(17,23)(21,24)(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=i,b^2=g,c^2=g,d^2=gi,e^2=gi,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, 8833962);