#461952

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16410
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

577 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:

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

Polycyclic presentation:

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

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, 369148);