#141104

order	512
nilpotency class	4
isoclinism	isoclinic class of #140973
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 322 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eghi,d^2=ghi,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48300);