#446276

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=efg,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353472);