#497141

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33716	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

646 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 186 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)(9,10,14,19,15,21,32,20)(11,16,25,37,23,35,51,38)(12,26,30,54,28,53,58,55)(13,31,27,18,33,43,56,44)(17,39,41,57,40,65,42,29)(22,45,48,66,49,68,70,67)(24,52,46,36,50,64,69,63)(34,59,60,71,62,72,61,47);(1,3,6,2)(4,7,5,8)(9,11,14,25,15,23,32,51)(10,16,19,37,21,35,20,38)(12,22,30,48,28,49,58,70)(13,24,27,46,33,50,56,69)(17,34,41,60,40,62,42,61)(18,36,43,64,44,63,31,52)(26,45,54,66,53,68,55,67)(29,47,39,59,57,71,65,72);(1,4,6,5)(2,7,3,8)(9,12,27,39,14,30,33,57,15,28,56,65,32,58,13,29)(10,17,31,55,20,42,44,53,21,40,43,54,19,41,18,26)(11,22,46,59,25,48,50,71,23,49,69,72,51,70,24,47)(16,34,52,67,38,61,63,68,35,62,64,66,37,60,36,45);(1,5,6,4)(2,7,3,8)(11,23)(12,28)(16,35)(17,40)(24,50)(25,51)(26,53)(29,57)(30,58)(36,63)(37,38)(39,65)(41,42)(46,69)(52,64)(54,55)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404337);