#93571

order	512
nilpotency class	3
isoclinism	isoclinic class of #93508
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,ℤ/4)
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×#405	C₂²
socle	C₂²	#1290
prime radical	C₂⁴	#108
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 200 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 — 2 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=f,e^2=f,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,h^a=hi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 767);