#453128

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16082
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

583 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)(3,15,20,9)(4,21,25,41)(5,11,28,38)(6,13)(7,14)(8,12)(10,24,36,29)(16,43)(17,31)(18,33,19,32)(22,30)(23,50)(26,39,27,40)(34,45)(35,54)(37,48)(42,49)(44,58,63,59)(46,56)(47,57)(51,61,64,55)(52,60)(53,62);(1,3)(2,9)(4,22,25,49)(5,17)(6,18)(7,19)(8,20)(10,34,36,43)(11,37)(12,15)(13,32)(14,33)(16,24,45,29)(21,42,41,30)(23,51,53,64)(26,46)(27,47)(28,48)(31,38)(35,59,60,58)(39,57)(40,56)(44,52,63,54)(50,55,62,61);(1,4)(2,10,12,36)(3,16)(5,23)(6,24)(7,29)(8,25)(9,30,15,42)(11,35,38,60)(13,21,14,41)(17,44)(18,22)(19,49)(20,45)(26,52)(27,54)(28,53)(31,55,37,61)(32,34,33,43)(39,50,40,62)(46,51)(47,64)(48,63)(56,58,57,59);(1,5,6,26,8,28,7,27)(2,11,14,40,12,38,13,39)(3,17,18,46,20,48,19,47)(4,23,24,52,25,53,29,54)(9,31,33,57,15,37,32,56)(10,35,41,62,36,60,21,50)(16,44,22,51,45,63,49,64)(30,55,43,59,42,61,34,58)

Polycyclic presentation:

2^9|a^2=efgh,b^2=fg,c^2=e,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,h^b=hi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360324);