#8932223

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	#45884	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gh,e^2=gi,f^2=h,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, 8839419);