#134676

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	#1808
prime radical	C₂⁴	#107
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)(12,27,30,17)(13,32)(14,35)(15,19)(16,24)(18,22)(20,33)(23,50,28,43)(25,56)(26,45)(29,61,63,64)(31,41,62,58)(34,65)(36,52,39,59)(37,55)(38,67)(40,48,57,51)(42,49)(44,47)(46,54)(53,69,60,72)(66,71,68,70);(9,11,22,47,14,25,34,55,15,26,49,67,33,54,13,24)(10,16,32,46,20,38,42,45,19,37,65,56,35,44,18,21)(12,28,48,71,31,60,63,52,30,23,51,70,62,53,29,59)(17,39,64,72,41,68,57,43,27,36,61,69,58,66,40,50);(9,12)(10,17)(11,23)(13,29)(14,31)(15,30)(16,36)(18,40)(19,27)(20,41)(21,43)(22,48)(24,52)(25,53)(26,28)(32,64)(33,62)(34,63)(35,58)(37,39)(38,66)(42,57)(44,68)(45,50)(46,69)(47,70)(49,51)(54,60)(55,59)(56,72)(61,65)(67,71)

Polycyclic presentation:

2^9|a^2=f,b^2=dg,d^2=gi,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^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, 41872);