#148194

order	512
nilpotency class	4
isoclinism	isoclinic class of #137661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

668 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)(5,6,10,17,11,18,32,15,12,16,33,47,31,46,9,29)(7,13,21,38,22,39,20,44,23,37,57,65,56,64,30,36)(8,24,26,41,27,45,60,49,28,48,61,14,34,42,25,43)(19,50,52,35,53,62,51,63,54,67,55,40,59,68,58,66);(1,2,3,4)(5,7)(6,13)(8,19)(9,30)(10,21)(11,22)(12,23)(14,40)(15,44)(16,37)(17,38)(18,39)(20,32)(24,50)(25,58)(26,52)(27,53)(28,54)(29,36)(31,56)(33,57)(34,59)(35,41)(42,68)(43,66)(45,62)(46,64)(47,65)(48,67)(49,63)(51,60)(55,61);(5,8,12,28)(6,14,16,41)(7,19,23,54)(9,26,32,61)(10,25,33,60)(11,34,31,27)(13,35,37,40)(15,45,29,42)(17,48,47,24)(18,49,46,43)(20,55,30,52)(21,58,57,51)(22,59,56,53)(36,62,44,68)(38,50,65,67)(39,66,64,63)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55390);