#147659

order	512
nilpotency class	4
isoclinism	isoclinic class of #13396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6)(7,18)(8,24,27,41)(9,28)(10,16)(11,34)(12,17)(13,20)(14,26,40,33)(15,31)(19,48,51,67)(21,42)(22,55)(23,47)(25,39,60,66)(29,44)(30,43)(32,45)(35,58,38,64)(36,52)(37,53)(46,50,65,54)(49,56,68,63)(57,62,61,59);(5,7,12,23)(6,13,17,37)(8,19,27,51)(9,29,32,52)(10,22,11,21)(14,38,40,35)(15,42,43,55)(16,44,34,36)(18,45,47,28)(20,30,53,31)(24,56,41,63)(25,58,60,64)(26,54,33,50)(39,48,66,67)(46,61,65,57)(49,62,68,59);(2,4)(5,8)(6,14)(7,19)(9,25)(10,33)(11,26)(12,27)(13,35)(15,39)(16,41)(17,40)(18,46)(20,49)(21,54)(22,50)(23,51)(24,34)(28,61)(29,58)(30,62)(31,59)(32,60)(36,63)(37,38)(42,67)(43,66)(44,56)(45,57)(47,65)(48,55)(52,64)(53,68)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,h^b=hi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54855);