#471208

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16774
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

660 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=f,d^2=ef,f^2=i,g^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,e^b=ei,f^b=fi,d^c=de,h^c=hi,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, 378404);