#134660

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1807
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41856);