#7715560

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

4030 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)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26)(27,31)(28,32)(29,36)(30,37)(33,42)(34,43)(35,38)(39,47)(40,48)(41,46)(44,50)(45,51)(49,53)(52,54)(55,56);(1,2,4,5)(3,6,7,8)(9,10,14,12)(11,15,19,17)(13,16,21,20)(18,22,24,23)(26,27)(28,38)(33,36)(34,37)(39,50)(40,51)(46,49)(52,56);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27,31,26)(28,32,38,35)(29,33,42,36)(30,37,43,34)(39,47,50,44)(40,45,51,48)(41,46,53,49)(52,55,56,54);(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,28,31,38)(26,32,27,35)(29,39,42,50)(30,40,43,51)(33,44,36,47)(34,45,37,48)(41,52,53,56)(46,54,49,55);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29,31,42)(26,33,27,36)(28,39,38,50)(30,41,43,53)(32,44,35,47)(34,46,37,49)(40,52,51,56)(45,54,48,55);(25,30,31,43)(26,34,27,37)(28,40,38,51)(29,41,42,53)(32,45,35,48)(33,46,36,49)(39,52,50,56)(44,54,47,55)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,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, 7622756);