#8932185

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=g,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, 8839381);