#94215

order	512
nilpotency class	5
isoclinism	isoclinic class of #94206
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

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	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

197 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,7,8,5)(4,6)(10,16,17,21)(11,12)(13,29)(15,30)(18,36,32,31)(19,38,34,37)(20,39,35,33)(22,25)(23,27)(24,40)(26,28);(1,3)(2,5)(4,8)(6,7)(9,10,12,18,25,34,24,39,30,35,40,38,29,36,11,21,14,17,27,32,13,19,26,33,15,20,28,37,22,31,23,16)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgh,e^2=fg,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,h^a=hi,c^b=ci,d^b=dg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1411);