#7659030

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

4068 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)(10,14)(11,17)(16,21)(18,23)(25,26,31,29)(27,35,38,32)(28,33,39,40)(30,34,43,42)(36,47,49,44)(37,48,51,45)(41,46,52,53)(50,55,56,54);(1,2,4,5)(3,6,7,8)(10,14)(15,20)(16,21)(22,24)(28,39)(33,40)(36,49)(41,52)(44,47)(46,53)(50,56)(54,55);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,31,38)(26,32,29,35)(28,36,39,49)(30,37,43,51)(33,44,40,47)(34,45,42,48)(41,50,52,56)(46,54,53,55);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28)(26,33)(27,36)(29,40)(30,41)(31,39)(32,44)(34,46)(35,47)(37,50)(38,49)(42,53)(43,52)(45,54)(48,55)(51,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,31,26)(27,35,38,32)(28,40,39,33)(30,34,43,42)(36,47,49,44)(37,45,51,48)(41,46,52,53)(50,54,56,55);(25,30)(26,34)(27,37)(28,41)(29,42)(31,43)(32,45)(33,46)(35,48)(36,50)(38,51)(39,52)(40,53)(44,54)(47,55)(49,56)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,e^2=h,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, 7566226);