#138394

order	512
nilpotency class	4
isoclinism	isoclinic class of #138028
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

535 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=gh,d^2=ghi,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,f^d=fi

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, 45590);