#149221

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2092
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 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:

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

Polycyclic presentation:

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

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, 56417);