#93068

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#331
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1296
prime radical	C₂³	#333
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

155 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 155 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

(2,6)(3,10)(4,11)(7,14)(8,9)(12,16)(17,18,21,26,23,29,35,28)(19,30,32,44,34,45,46,39)(20,27,22,25,36,42,37,43)(24,33,38,47,41,48,40,31);(1,2,3,7,11,9,16,15,5,8,12,14,4,6,10,13)(17,19,22,33,21,32,36,47,23,34,37,48,35,46,20,31)(18,24,27,39,28,40,43,45,29,41,42,44,26,38,25,30)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=h,e^2=fgi,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264);