#7715545

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

4058 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)(12,20)(15,17)(18,24)(25,26,30,31)(27,32,35,36)(28,39,41,33)(29,42,43,34)(37,49,47,44)(38,50,48,45)(40,46,53,52)(51,54,55,56);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(26,31)(27,35)(33,39)(34,42)(37,47)(38,48)(46,52)(51,55);(1,3,4,7)(2,6,5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,36)(28,39)(29,34)(30,31)(32,35)(33,41)(37,44)(38,50)(40,52)(42,43)(45,48)(46,53)(47,49)(51,54)(55,56);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,32)(28,37)(29,38)(30,35)(31,36)(33,44)(34,45)(39,49)(40,51)(41,47)(42,50)(43,48)(46,54)(52,56)(53,55);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,28,30,41)(26,33,31,39)(27,37,35,47)(29,40,43,53)(32,44,36,49)(34,46,42,52)(38,51,48,55)(45,54,50,56);(25,29,30,43)(26,34,31,42)(27,38,35,48)(28,40,41,53)(32,45,36,50)(33,46,39,52)(37,51,47,55)(44,54,49,56)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,e^2=h,f^2=i,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, 7622741);