#404117

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311313);