#453941

order	512
nilpotency class	3
isoclinism	isoclinic class of #453940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16245
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)(3,13,16,8)(4,18,22,9)(5,24,25,10)(6,27,20,11)(14,30,42,39)(15,40,43,31)(17,32,28,37)(19,46,50,33)(21,47,54,34)(23,35,51,48)(26,36,29,38)(41,63,64,56)(44,57,45,58)(49,59,52,60)(53,61,55,62);(1,3,6,28,7,16,20,17)(2,8,11,37,12,13,27,32)(4,19,23,14,22,50,51,42)(5,15,26,45,25,43,29,44)(9,33,35,30,18,46,48,39)(10,31,36,58,24,40,38,57)(21,52,53,41,54,49,55,64)(34,60,61,56,47,59,62,63);(1,4,20,51,7,22,6,23)(2,9,27,48,12,18,11,35)(3,14,28,50,16,42,17,19)(5,21,26,53,25,54,29,55)(8,30,37,46,13,39,32,33)(10,34,36,61,24,47,38,62)(15,41,44,52,43,64,45,49)(31,56,57,60,40,63,58,59);(1,5)(2,10)(3,15,16,43)(4,21)(6,29)(7,25)(8,31,13,40)(9,34)(11,38)(12,24)(14,41,42,64)(17,45,28,44)(18,47)(19,49,50,52)(20,26)(22,54)(23,55)(27,36)(30,56,39,63)(32,58,37,57)(33,59,46,60)(35,62)(48,61)(51,53)

Polycyclic presentation:

2^9|a^2=i,b^2=efg,c^2=egh,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,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, 361137);