#134981

order	512
nilpotency class	4
isoclinism	isoclinic class of #134965
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

256 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,7)(6,8)(9,10)(11,21)(12,27,29,17)(13,31)(14,34)(15,19)(16,24)(18,22)(20,32)(23,48,50,41)(25,54)(26,43)(28,58,60,61)(30,39,59,57)(33,62)(35,49,55,71)(36,53)(37,64)(38,46,56,68)(40,47)(42,45)(44,52)(51,66,70,69)(63,72,65,67);(9,11,22,45,14,25,33,53,15,26,47,64,32,52,13,24)(10,16,31,44,20,37,40,43,19,36,62,54,34,42,18,21)(12,23,46,67,30,51,60,71,29,50,68,72,59,70,28,49)(17,35,61,66,39,63,56,48,27,55,58,69,57,65,38,41);(1,2,4,6)(3,8,7,5)(9,12,15,29)(10,17,19,27)(11,23,26,50)(13,28,33,60)(14,30,32,59)(16,35,36,55)(18,38,40,56)(20,39,34,57)(21,41,43,48)(22,46,47,68)(24,49,53,71)(25,51,52,70)(31,61,62,58)(37,63,42,65)(44,66,54,69)(45,67,64,72)

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=h,d^2=gi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^c=ehi,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42177);