#150823

order	512
nilpotency class	5
isoclinism	isoclinic class of #150397
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

295 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,7)(4,6)(5,8)(9,10,13,19,15,21,35,51)(11,22,24,48,26,42,58,66)(12,27,30,49,29,17,33,46)(14,20,38,50,39,18,34,47)(16,32,41,36,43,57,67,25)(23,53,55,71,54,44,56,68)(28,40,60,63,59,64,72,65)(31,52,62,70,37,45,61,69);(2,6)(3,4)(7,8)(9,11,15,26)(10,16,21,43)(12,28,29,59)(13,32,35,57)(14,36,39,25)(17,44,27,53)(18,42,20,22)(19,48,51,66)(23,37,54,31)(24,34,58,38)(30,56,33,55)(40,52,64,45)(41,50,67,47)(46,63,49,65)(60,61,72,62)(68,69,71,70);(1,3,5,4)(2,6,8,7)(9,12)(10,17)(11,23)(13,33)(14,37)(15,29)(16,40)(18,45)(19,49)(20,52)(21,27)(22,44)(24,56)(25,59)(26,54)(28,36)(30,35)(31,39)(32,60)(34,61)(38,62)(41,65)(42,53)(43,64)(46,51)(47,70)(48,71)(50,69)(55,58)(57,72)(63,67)(66,68)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,d^c=dh,e^c=ei,f^c=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58019);