#93510

order	512
nilpotency class	3
isoclinism	isoclinic class of #93508
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,ℤ/4)
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂²×#131	C₂²
socle	C₂²	#1289
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

454 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,5)(4,7)(6,8)(9,10,13,19,16,21,36,52)(11,22,24,54,27,56,33,66)(12,28,30,62,32,47,34,67)(14,37,39,50,15,20,35,51)(17,40,41,57,43,58,70,23)(18,44,46,31,48,65,49,38)(25,55,61,72,26,45,59,71)(29,42,63,69,60,53,64,68);(2,5,7,6)(9,11,14,25,16,27,15,26)(10,17,20,42,21,43,37,53)(12,29,31,57,32,60,38,23)(13,33,35,61,36,24,39,59)(18,45,47,22,48,55,28,56)(19,41,50,68,52,70,51,69)(30,64,44,58,34,63,65,40)(46,71,62,66,49,72,67,54)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=i,e^2=i,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, 706);