#479754

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

722 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,4,6)(3,5,7,8)(9,10,16,20,15,22,14,21)(11,23,26,17,28,42,27,43)(12,29,39,47,33,46,32,18)(13,34,38,50,37,19,36,49)(24,40,30,44,55,60,54,51)(25,52,57,65,59,69,58,41)(31,48,64,70,63,61,35,45)(53,67,56,71,66,72,62,68);(1,3)(2,5)(4,7)(6,8)(9,11,14,27,15,28,16,26)(10,17,21,23,22,43,20,42)(12,30,32,24,33,54,39,55)(13,25,36,58,37,59,38,57)(18,44,46,40,47,51,29,60)(19,41,50,69,34,65,49,52)(31,62,35,66,63,56,64,53)(45,68,61,72,70,71,48,67);(2,6)(5,8)(9,12,14,32,15,33,16,39)(10,18,20,29,22,47,21,46)(11,24,26,30,28,55,27,54)(13,35,36,63,37,64,38,31)(17,40,23,51,43,60,42,44)(19,48,49,70,34,61,50,45)(25,56,57,66,59,62,58,53)(41,68,69,72,65,71,52,67);(3,7)(5,8)(9,13,15,37)(10,19,22,34)(11,25,28,59)(12,31,33,63)(14,38,16,36)(17,41,43,65)(18,45,47,70)(20,50,21,49)(23,52,42,69)(24,53,55,66)(26,58,27,57)(29,61,46,48)(30,62,54,56)(32,64,39,35)(40,67,60,72)(44,68,51,71)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386950);