#497048

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22209
prime radical	#575	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

449 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224962, #72320755
maximal normal subgroup of	#72224962, #72320755

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404244);