#94344

order	512
nilpotency class	5
isoclinism	isoclinic class of #94343
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1380
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂³×C₁₆	C₂²
prime radical	C₂⁴	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

231 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

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

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, 1540);