#135041

order	512
nilpotency class	4
isoclinism	isoclinic class of #135037
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#463
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

272 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,c^2=i,d^2=gi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=di,e^c=eh,h^c=hi

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, 42237);