#181248

order	512
nilpotency class	3
isoclinism	isoclinic class of #181129
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1677 conjugacy classes of subgroups; only the 868 normal subgroups are shown.

normal subgroups only

This group has 1677 conjugacy classes of subgroups — too many to draw. Only its 868 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=hi,g^2=hi,b^a=be,e^a=eh,g^a=ghi,f^b=fi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88444);