#128218

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1746
prime radical	C₂³	#405
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,e^2=g,f^2=h,g^2=i,b^a=bd,f^a=fi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35414);