#7659039

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

3974 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566235);