#7714749

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

4014 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621945);