#448228

order	512
nilpotency class	3
isoclinism	isoclinic class of #419458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2709
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15907
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

680 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efg,c^2=efg,d^2=ef,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^b=de,e^b=ei,f^b=fi,h^c=hi,e^d=ei,f^d=fi

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, 355424);