#93619

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	C₄×#97	C₂²
socle	C₂²	#1288
prime radical	C₂³	#339
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

182 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

(2,5)(3,7)(9,10,13,17,14,18,12,24,15,19,27,35,28,34,26,38)(11,20,21,32,22,33,25,39,23,36,37,16,29,30,40,31);(1,2,3,6,4,5,7,8)(9,11)(10,16)(12,25,26,40)(13,21,27,37)(14,22)(15,23)(17,33,35,30)(18,31)(19,32)(20,24,36,38)(28,29)(34,39)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=f,e^2=fgi,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, 815);