#148808

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#2065
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 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)(5,6,9,18,12,19,35,15)(7,20,22,13,26,44,25,45)(8,27,32,48,33,52,29,64)(10,36,11,16,34,17,40,51)(14,31,47,55,49,38,50,30)(21,46,58,66,59,68,56,67)(23,42,37,65,39,54,24,43)(28,62,57,53,60,41,63,61);(5,7,9,22,12,26,35,25)(6,13,15,20,19,45,18,44)(8,28,29,63,33,60,32,57)(10,37,11,39,34,24,40,23)(14,46,47,66,49,68,50,67)(16,43,36,54,51,65,17,42)(21,55,56,31,59,30,58,38)(27,61,48,62,52,53,64,41);(2,4)(5,8,12,33)(6,14,19,49)(7,21)(9,32,35,29)(10,38,34,31)(11,30,40,55)(13,41)(15,50,18,47)(16,48,51,64)(17,52,36,27)(20,53)(22,58)(23,60)(24,57)(25,56)(26,59)(28,39)(37,63)(42,46)(43,67)(44,61)(45,62)(54,68)(65,66)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fh,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, 56004);