#135989

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1852
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

413 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,e^b=eh,g^b=gi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43185);