#124788

order	512
nilpotency class	4
isoclinism	isoclinic class of #124708
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,5)(4,7)(6,8)(9,10,13,20,15,19,38,48,16,22,36,52,14,21,37,51)(11,23,25,54,27,43,34,66,28,42,61,49,26,17,39,41)(12,29,31,63,33,47,65,50,30,45,35,68,32,18,44,46)(24,55,57,70,59,62,64,71,56,40,60,69,58,53,67,72);(2,5)(3,8)(6,7)(9,11,16,28)(10,17,22,43)(12,24,30,56)(13,34,36,39)(14,27,15,26)(18,40,47,55)(19,42,21,23)(20,49,52,54)(25,37,61,38)(29,62,45,53)(31,64,35,67)(32,59,33,58)(41,48,66,51)(44,60,65,57)(46,71,50,72)(63,70,68,69);(9,12)(10,18,22,47)(11,24)(13,35)(14,32)(15,33)(16,30)(17,40,43,55)(19,29,21,45)(20,50,52,46)(23,53,42,62)(25,60)(26,58)(27,59)(28,56)(31,36)(34,67)(37,65)(38,44)(39,64)(41,70,66,69)(48,68,51,63)(49,72,54,71)(57,61)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fgh,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ch,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31984);