#450836

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15979
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

931 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,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,f^a=fi,c^b=cg,d^b=de,g^b=gi,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, 358032);