#460621

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

937 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)(3,5)(4,6)(7,8)(9,10,16,20,15,22,14,21)(11,23,39,43,27,42,26,17)(12,28,37,18,32,45,31,46)(13,33,38,49,36,19,35,48)(24,52,58,50,29,40,54,68)(25,51,59,41,57,70,56,64)(30,61,63,44,62,47,34,65)(53,71,66,72,60,67,55,69);(1,3,4,7)(2,5,6,8)(9,11,15,27)(10,17)(12,29,32,24)(13,25,36,57)(14,26,16,39)(18,40)(19,41)(20,23)(21,42)(22,43)(28,50)(30,60,62,53)(31,58,37,54)(33,64)(34,66,63,55)(35,56,38,59)(44,67)(45,68)(46,52)(47,69)(48,70)(49,51)(61,72)(65,71);(2,6)(5,8)(9,12,15,32)(10,18,22,46)(11,24,27,29)(13,34,36,63)(14,37,16,31)(17,40,43,52)(19,47,33,61)(20,28,21,45)(23,50,42,68)(25,55,57,66)(26,58,39,54)(30,38,62,35)(41,69,64,72)(44,49,65,48)(51,71,70,67)(53,59,60,56);(9,13,15,36)(10,19,22,33)(11,25,27,57)(12,30,32,62)(14,38,16,35)(17,41,43,64)(18,44,46,65)(20,49,21,48)(23,51,42,70)(24,53,29,60)(26,59,39,56)(28,47,45,61)(31,63,37,34)(40,67,52,71)(50,69,68,72)(54,66,58,55)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367817);