#93088

order	512
nilpotency class	4
isoclinism	isoclinic class of #93083
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 150 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=h,e^2=fgh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284);