#6443735

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439035
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 72 points:

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

Polycyclic presentation:

2^9|c^2=g,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

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, 6350931);