#477005

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17101
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1035 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384201);