#93180

order	512
nilpotency class	4
isoclinism	isoclinic class of #93165
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×C₂²⋊Q₈	C₂²
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₂²⋊Q₈ ▷ C₂⁴ ▷ C₂² ▷ C₁

481 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 48 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=hi,e^2=hi,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=gi,c^b=cg,d^b=dfi,f^b=fi,g^b=gi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376);