#149460

order	512
nilpotency class	4
isoclinism	isoclinic class of #129558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1786
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

684 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)(11,21)(12,26,28,17)(13,30)(14,36)(15,19)(16,23)(18,34)(20,33)(22,47,50,68)(24,48)(25,54)(27,57,32,65)(29,42,59,56)(31,46)(35,62)(37,43)(38,49,55,52)(39,64)(40,53)(41,60,44,66)(45,67)(51,70,61,72)(58,69,63,71);(1,2,4,6)(3,7,8,5)(9,11)(10,16)(12,22)(13,31)(14,37)(15,24)(17,38)(18,43)(19,40)(20,46)(21,30)(23,34)(25,33)(26,55)(27,58)(28,50)(29,61)(32,63)(35,64)(36,39)(41,70)(42,71)(44,72)(45,54)(47,57)(48,62)(49,60)(51,59)(52,66)(53,67)(56,69)(65,68);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,22)(13,32)(14,29)(15,28)(16,38)(18,44)(19,26)(20,42)(21,47)(23,52)(24,50)(25,51)(27,35)(30,57)(31,63)(33,59)(34,66)(36,56)(37,61)(39,69)(40,55)(41,45)(43,72)(46,71)(48,68)(49,53)(54,70)(58,64)(60,67)(62,65)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,d^b=dh,e^b=ei,h^b=hi,d^c=di,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, 56656);