#497121

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

588 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 218 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)(3,6)(4,7)(5,8)(9,10,14,20,15,19,34,43,16,21,35,44,13,32,33,58)(11,17,25,39,26,38,53,63,23,37,50,62,24,51,52,69)(12,27,30,55,31,41,57,64,28,54,56,70,29,18,40,42)(22,45,47,65,48,61,67,71,49,66,68,72,46,36,59,60);(1,3,6,2)(4,7,5,8)(9,11,14,25,15,26,34,53,16,23,35,50,13,24,33,52)(10,17,20,39,19,38,43,63,21,37,44,62,32,51,58,69)(12,22,30,47,31,48,57,67,28,49,56,68,29,46,40,59)(18,36,42,60,27,45,55,65,41,61,64,71,54,66,70,72);(1,4,6,5)(2,7,3,8)(9,12,13,29,16,28,15,31)(10,18,19,27,21,41,32,54)(11,22,24,46,23,49,26,48)(14,30,33,40,35,56,34,57)(17,36,38,45,37,61,51,66)(20,42,43,55,44,64,58,70)(25,47,52,59,50,68,53,67)(39,60,63,65,62,71,69,72);(1,5,6,4)(2,7,3,8)(11,23)(12,28)(17,37)(18,41)(24,26)(25,50)(27,54)(29,31)(30,56)(38,51)(39,62)(40,57)(42,64)(52,53)(55,70)(63,69)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404317);