#463901

order	512
nilpotency class	3
isoclinism	isoclinic class of #459490
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2733
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16495
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

807 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371097);