#150081

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

220 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=fi,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df,d^b=dfi,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57277);