#7714847

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

4003 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,e^2=hi,f^2=h,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, 7622043);