#453879

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16192
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

527 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=eg,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 361075);