#412444

order	512
nilpotency class	3
isoclinism	isoclinic class of #400512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2491
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15130
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1345 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)(3,15,18,9,19,33,20,34)(4,21,24,37,25,36,30,10)(5,26,27,39,29,38,28,11)(16,42,44,57,22,31,48,40)(17,32,45,53,46,58,47,41)(23,35,51,49,52,59,55,54)(43,60,62,61,50,64,63,56);(1,3,6,18,8,19,7,20)(2,9,12,15,14,34,13,33)(4,22,24,48,25,16,30,44)(5,17,27,45,29,46,28,47)(10,31,36,57,37,42,21,40)(11,32,38,41,39,58,26,53)(23,50,51,63,52,43,55,62)(35,56,54,64,59,61,49,60);(1,4)(2,10)(3,16)(5,23)(6,30)(7,24)(8,25)(9,31)(11,35)(12,21)(13,36)(14,37)(15,40)(17,43)(18,48)(19,22)(20,44)(26,54)(27,55)(28,51)(29,52)(32,56)(33,57)(34,42)(38,49)(39,59)(41,60)(45,63)(46,50)(47,62)(53,64)(58,61);(1,5,7,28,8,29,6,27)(2,11,13,26,14,39,12,38)(3,17,20,47,19,46,18,45)(4,23,24,51,25,52,30,55)(9,32,33,53,34,58,15,41)(10,35,36,54,37,59,21,49)(16,43,44,62,22,50,48,63)(31,56,57,64,42,61,40,60)

Polycyclic presentation:

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

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, 319640);