#134659

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

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,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, 41855);