#7715469

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

4065 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)(11,15)(12,19)(13,21)(14,17)(16,22)(18,24)(20,23)(25,26,31,32)(27,29,36,33)(28,34,39,46)(30,44,45,35)(37,41,40,47)(38,48,50,43)(42,55,52,49)(51,56,53,54);(1,2,4,5)(3,6,7,8)(9,11,14,12)(10,15,17,19)(13,18,22,20)(16,23,21,24)(26,32)(27,36)(28,39)(35,44)(38,50)(41,47)(42,52)(54,56);(9,12,14,11)(10,15,17,19)(13,20,22,18)(16,23,21,24)(25,27,31,36)(26,33,32,29)(28,40,39,37)(30,38,45,50)(34,41,46,47)(35,48,44,43)(42,53,52,51)(49,54,55,56);(9,10)(11,15)(12,19)(13,16)(14,17)(18,23)(20,24)(21,22)(25,28)(26,34)(27,37)(29,41)(30,42)(31,39)(32,46)(33,47)(35,49)(36,40)(38,51)(43,54)(44,55)(45,52)(48,56)(50,53);(1,3,4,7)(2,6,5,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,29,31,33)(26,27,32,36)(28,41,39,47)(30,43,45,48)(34,37,46,40)(35,38,44,50)(42,54,52,56)(49,51,55,53);(25,30,31,45)(26,35,32,44)(27,38,36,50)(28,42,39,52)(29,43,33,48)(34,49,46,55)(37,51,40,53)(41,54,47,56)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,e^2=g,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, 7622665);