#9039150

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	#55291	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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)(9,10,14,15)(11,17,19,12)(13,21,22,16)(18,20,24,23)(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)(10,15)(12,17)(16,21)(20,23)(27,36)(32,35)(37,50)(38,43)(45,49)(46,48)(51,53)(55,56);(1,2,5,4)(3,6,7,8)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,27,31,36)(26,32,29,35)(28,37,41,50)(30,43,44,38)(33,45,39,49)(34,48,42,46)(40,53,54,51)(47,56,52,55);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24)(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);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,29,31,26)(27,32,36,35)(28,33,41,39)(30,42,44,34)(37,49,50,45)(38,46,43,48)(40,47,54,52)(51,56,53,55);(1,4,5,2)(3,8,7,6)(9,13,14,22)(10,16,15,21)(11,18,19,24)(12,20,17,23)(25,30,31,44)(26,34,29,42)(27,38,36,43)(28,40,41,54)(32,46,35,48)(33,47,39,52)(37,51,50,53)(45,55,49,56)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=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, 8946346);