#8932532

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	#56366	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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)(11,17)(14,18)(19,23)(21,24)(25,26,31,35)(27,32,36,29)(28,39,43,33)(30,34,47,50)(37,44,40,41)(38,48,46,45)(42,52,55,49)(51,56,53,54);(2,6)(3,8)(9,10,15,12)(11,14,17,18)(13,20,22,16)(19,24,23,21)(27,36)(29,32)(37,40)(38,46)(41,44)(45,48)(51,53)(54,56);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,32)(28,40)(29,35)(30,46)(31,36)(33,44)(34,45)(37,43)(38,47)(39,41)(42,51)(48,50)(49,54)(52,56)(53,55);(1,4)(2,3)(5,7)(6,8)(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,24,23,21)(25,28,31,43)(26,33,35,39)(27,37,36,40)(29,44,32,41)(30,42,47,55)(34,49,50,52)(38,51,46,53)(45,56,48,54);(1,2)(3,4)(5,6)(7,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,31,32)(26,27,35,36)(28,41,43,44)(30,45,47,48)(33,40,39,37)(34,38,50,46)(42,54,55,56)(49,53,52,51);(1,4)(2,3)(5,7)(6,8)(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,30)(26,34)(27,38)(28,42)(29,45)(31,47)(32,48)(33,49)(35,50)(36,46)(37,51)(39,52)(40,53)(41,54)(43,55)(44,56)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839728);