#149466

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1786
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

678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(9,10)(11,22)(12,27,30,45)(13,31)(14,20)(15,37)(16,21)(17,24)(18,29,44,36)(19,34)(23,51,54,71)(25,49)(26,50)(28,43,61,69)(32,40)(33,47)(35,63)(38,46)(39,66,67,59)(41,64)(42,56)(48,53,70,55)(52,57,72,68)(58,65,62,60);(1,2,4,6)(3,7,8,5)(9,11)(10,17)(12,23)(13,32)(14,25)(15,38)(16,26)(18,39)(19,46)(20,40)(21,42)(22,31)(24,34)(27,57)(28,59)(29,53)(30,54)(33,56)(35,64)(36,55)(37,41)(43,51)(44,67)(45,68)(47,49)(48,62)(50,63)(52,60)(58,70)(61,66)(65,72)(69,71);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,23)(13,28)(14,36)(15,29)(16,30)(17,39)(19,43)(20,45)(21,44)(22,48)(24,52)(25,55)(26,54)(27,37)(31,62)(32,59)(33,65)(34,60)(35,61)(38,53)(40,68)(41,57)(42,67)(46,51)(47,69)(49,71)(50,70)(56,72)(58,63)(64,66)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=hi,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,h^b=hi,e^c=ei,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, 56662);