#147681

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
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

992 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,d^b=dhi,f^b=fi,g^b=gi,h^b=hi,f^c=fi,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, 54877);