#475497

order	512
nilpotency class	3
isoclinism	isoclinic class of #469927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2605
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17055
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

533 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

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, 382693);