#471211

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16779
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

658 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:

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

Polycyclic presentation:

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

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, 378407);