#148196

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1893
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

676 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)(5,6,11,18,12,19,35,49,13,20,34,48,10,17,9,16)(7,14,24,40,23,39,56,65,25,41,58,67,36,50,22,38)(8,26,30,46,31,47,28,43,32,60,33,15,29,44,61,45)(21,51,53,64,27,59,52,66,54,68,55,37,57,63,62,42);(1,2,3,4)(5,7)(6,14)(8,27)(9,22)(10,36)(11,24)(12,23)(13,25)(15,42)(16,38)(17,50)(18,40)(19,39)(20,41)(21,29)(26,59)(28,55)(30,52)(31,54)(32,57)(33,62)(34,58)(35,56)(37,43)(44,51)(45,64)(46,66)(47,68)(48,67)(49,65)(53,61)(60,63);(5,8)(6,15)(7,21)(9,33)(10,31)(11,28)(12,29)(13,32)(14,37)(16,47)(17,45)(18,26)(19,43)(20,46)(22,55)(23,57)(24,52)(25,54)(27,36)(30,35)(34,61)(38,59)(39,66)(40,51)(41,64)(42,50)(44,49)(48,60)(53,56)(58,62)(63,65)(67,68)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=gh,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, 55392);