#480051

order	512
nilpotency class	4
isoclinism	isoclinic class of #33348
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1009 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1009 conjugacy classes of subgroups — too many to draw. Only its 314 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,e^b=ei,f^b=fg,d^c=di,f^e=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387247);