#150483

order	512
nilpotency class	5
isoclinism	isoclinic class of #150397
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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

299 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 54 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)(9,10,13,19,15,21,33,49)(11,22,24,47,26,40,54,64)(12,17,29,44,28,45,58,70)(14,20,36,48,37,18,32,46)(16,31,39,34,41,53,65,25)(23,42,52,67,51,66,59,72)(27,56,57,71,55,38,62,63)(30,43,61,68,35,50,60,69);(2,6)(3,4)(7,8)(9,11,15,26)(10,16,21,41)(12,27,28,55)(13,31,33,53)(14,34,37,25)(17,42,45,66)(18,40,20,22)(19,47,49,64)(23,35,51,30)(24,32,54,36)(29,59,58,52)(38,50,56,43)(39,48,65,46)(44,63,70,71)(57,60,62,61)(67,68,72,69);(1,3,5,4)(2,7,8,6)(9,12)(10,17)(11,23)(13,29)(14,35)(15,28)(16,38)(18,43)(19,44)(20,50)(21,45)(22,42)(24,52)(25,55)(26,51)(27,34)(30,37)(31,62)(32,61)(33,58)(36,60)(39,63)(40,66)(41,56)(46,68)(47,67)(48,69)(49,70)(53,57)(54,59)(64,72)(65,71)

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,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,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, 57679);