#140616

order	512
nilpotency class	4
isoclinism	isoclinic class of #13229
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 104 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 — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ei,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,f^b=fg,g^b=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47812);