#148048

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1874
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

700 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,e^c=eh,f^c=fi,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, 55244);