#499384

order	512
nilpotency class	4
isoclinism	isoclinic class of #13497
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2075
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2075
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320541, #72320647, #72320665
maximal normal subgroup of	#72320541, #72320647, #72320665
maximal quotient of	#72322813

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406580);