#150734

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	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2310 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 380 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)(3,7,8,5)(4,6)(9,10,13,19,15,21,34,52)(11,22,24,50,26,43,58,68)(12,27,29,51,31,17,33,47)(14,18,38,49,39,20,35,53)(16,40,42,25,44,32,64,36)(23,54,55,72,56,45,57,69)(28,60,61,67,59,41,65,66)(30,48,63,71,37,46,62,70);(2,5)(6,7)(9,11,15,26)(10,16,21,44)(12,28,31,59)(13,32,34,40)(14,36,39,25)(17,45,27,54)(18,43,20,22)(19,50,52,68)(23,37,56,30)(24,35,58,38)(29,57,33,55)(41,46,60,48)(42,53,64,49)(47,66,51,67)(61,62,65,63)(69,71,72,70);(3,8)(5,7)(9,12,15,31)(10,17,21,27)(11,23,26,56)(13,33,34,29)(14,37,39,30)(16,41,44,60)(18,48,20,46)(19,51,52,47)(22,45,43,54)(24,57,58,55)(25,59,36,28)(32,61,40,65)(35,62,38,63)(42,67,64,66)(49,70,53,71)(50,72,68,69)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,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, 57930);