#94247

order	512
nilpotency class	5
isoclinism	isoclinic class of #94180
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1393
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₄×C₄⋊Q₈	C₂²
prime radical	C₂³	#342
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄×C₄⋊Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 225 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=i,e^2=fgh,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=ghi,h^a=hi,c^b=ci,d^b=dg,d^c=dhi,e^d=ei,f^d=fi,g^d=gi

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, 1443);