#483595

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#442
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390791);