#94912

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

576 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=chi,d^a=dg,f^a=fg,c^b=cgi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240037
maximal normal subgroup of	#72240037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2108);