#480015

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
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	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1571 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1571 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,e^b=ei,f^b=fgi,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, 387211);