#148514

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	#2054
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

654 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)(7,20)(8,26)(9,33)(10,36)(11,16)(12,38)(13,19)(14,22)(15,29)(17,35)(18,34)(21,40)(23,50)(24,42)(25,49)(27,51)(28,59)(30,48)(31,62)(32,56)(37,44)(39,41)(43,55)(45,57)(46,60)(47,61)(52,64)(53,67)(54,66)(58,68)(63,65);(5,7)(6,14)(8,27)(9,23)(10,37)(11,22)(12,39)(13,25)(15,45)(16,41)(17,49)(18,50)(19,44)(20,33)(21,30)(24,34)(26,40)(28,53)(29,54)(31,52)(32,58)(35,55)(36,42)(38,43)(46,67)(47,68)(48,64)(51,59)(56,66)(57,60)(61,63)(62,65);(5,8,9,28,34,60,10,29,13,32,35,61,12,31,11,30)(6,15,16,46,38,59,17,26,19,48,36,62,18,47,33,56)(7,21,22,52,39,63,55,58,25,54,37,57,24,53,23,27)(14,40,20,51,50,67,42,45,44,66,49,68,43,65,41,64)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=hi,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fhi,h^b=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, 55710);