#509108

order	512
nilpotency class	4
isoclinism	isoclinic class of #509107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30396
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24288	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2932 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320520, #72320582, #72320602
maximal normal subgroup of	#72320520, #72320582, #72320602
maximal quotient of	#72321415, #72321598, #72321674, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416304);