#453922

order	512
nilpotency class	3
isoclinism	isoclinic class of #427841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2778
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16225
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

545 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=efg,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 361118);