#8932524

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=gi,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, 8839720);