#453899

order	512
nilpotency class	3
isoclinism	isoclinic class of #453898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16208
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

639 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361095);