#6439609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
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	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2219 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,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, 6346805);