#480013

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	C₂³×#41	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1624 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1624 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)(11,21)(12,17)(13,18)(14,28,29,19)(15,20)(16,23,31,27)(22,32)(24,37,38,33)(25,34)(26,35)(30,36,40,39);(9,11,15,13)(10,16,20,31)(12,22,25,26)(14,23,29,27)(17,30,34,40)(18,19,21,28)(24,36,38,39)(32,37,35,33);(9,12,15,25)(10,17,20,34)(11,22,13,26)(14,24,29,38)(16,30,31,40)(18,35,21,32)(19,33,28,37)(23,36,27,39);(9,13,15,11)(10,18,20,21)(12,22,25,26)(14,27,29,23)(16,19,31,28)(17,32,34,35)(24,36,38,39)(30,37,40,33)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,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, 387209);