#149012

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	#2073
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

436 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:

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

Polycyclic presentation:

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

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, 56208);