#396794

order	512
nilpotency class	3
isoclinism	isoclinic class of #394481
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2472
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14738
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

939 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303990);