#8417336

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	#52653	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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)(3,7)(4,8)(5,6)(10,12)(15,18)(16,20)(22,24)(27,35)(31,36)(37,40)(38,49)(45,47)(46,50)(51,53)(55,56);(2,6)(3,8)(11,17)(15,18)(19,23)(22,24)(25,26,30,34)(27,36,35,31)(28,39,42,32)(29,33,44,43)(37,45,40,47)(38,50,49,46)(41,52,54,48)(51,55,53,56);(1,3)(2,4)(5,8)(6,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,26)(27,31)(28,32)(29,43)(30,34)(33,44)(35,36)(37,45)(38,50)(39,42)(40,47)(41,52)(46,49)(48,54)(51,56)(53,55);(1,3)(2,4)(5,8)(6,7)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,27,30,35)(26,31,34,36)(28,40,42,37)(29,38,44,49)(32,47,39,45)(33,46,43,50)(41,53,54,51)(48,56,52,55);(1,4)(2,3)(5,7)(6,8)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,28,30,42)(26,32,34,39)(27,37,35,40)(29,41,44,54)(31,45,36,47)(33,48,43,52)(38,51,49,53)(46,55,50,56);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,29)(26,33)(27,38)(28,41)(30,44)(31,46)(32,48)(34,43)(35,49)(36,50)(37,51)(39,52)(40,53)(42,54)(45,55)(47,56)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324532);