#453127

order	512
nilpotency class	3
isoclinism	isoclinic class of #424595
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	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

577 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:

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

Polycyclic presentation:

2^9|a^2=efgh,b^2=fg,c^2=e,d^2=ef,f^2=i,g^2=i,h^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

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, 360323);