#148649

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2076
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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)(7,18,22,37)(8,24,29,59)(9,14)(10,32)(11,17)(12,20,38,23)(13,27,41,63)(15,30)(16,31)(19,48)(21,53,55,36)(25,60)(26,58,28,44)(33,42,62,40)(34,43)(35,54)(39,50)(45,61)(46,64)(47,52,67,56)(49,65)(51,57)(66,68);(1,2)(3,4)(5,7)(6,12)(8,25)(9,23)(10,21)(11,22)(13,39)(14,37)(15,36)(16,18)(17,38)(19,28)(20,31)(24,57)(26,51)(27,54)(29,49)(30,56)(32,47)(33,64)(34,55)(35,40)(41,46)(42,66)(43,67)(44,60)(45,53)(48,59)(50,62)(52,61)(58,65)(63,68);(2,4)(5,8,11,29)(6,13)(7,19)(9,28,31,26)(10,33,34,62)(12,35,38,66)(14,42)(15,44)(16,40)(17,41)(18,46,37,39)(20,25)(21,54)(22,51)(23,49)(24,32)(27,61,63,30)(36,57,53,48)(43,59)(45,58)(47,65,67,60)(50,56)(52,64)(55,68)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55845);