#148192

order	512
nilpotency class	4
isoclinism	isoclinic class of #137659
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×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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)(5,6,10,17,30,45,31,46,12,16,9,15,11,18,33,48)(7,13,20,36,55,63,60,68,22,37,28,43,21,38,56,64)(8,23,25,42,34,47,29,14,27,41,24,49,26,44,32,40)(19,50,51,65,57,67,59,39,53,66,58,61,52,62,54,35);(1,2,3,4)(5,7)(6,13)(8,19)(9,28)(10,20)(11,21)(12,22)(14,39)(15,43)(16,37)(17,36)(18,38)(23,50)(24,58)(25,51)(26,52)(27,53)(29,59)(30,55)(31,60)(32,54)(33,56)(34,57)(35,40)(41,66)(42,65)(44,62)(45,63)(46,68)(47,67)(48,64)(49,61);(5,8,12,27)(6,14,16,40)(7,19,22,53)(9,29,10,32)(11,34,30,26)(13,35,37,39)(15,44,17,47)(18,49,45,42)(20,54,28,59)(21,57,55,52)(23,48,41,46)(24,33,25,31)(36,62,43,67)(38,65,63,61)(50,68,66,64)(51,60,58,56)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,e^b=ei,e^c=eh,g^c=gh,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, 55388);