#8754896

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8662092);