#375669

order	512
nilpotency class	3
isoclinism	isoclinic class of #213728
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1159 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1159 conjugacy classes of subgroups — too many to draw. Only its 250 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)(3,7)(4,5)(6,8)(9,10,14,19,15,12,17,25)(11,20,22,33,23,16,28,31)(13,24,27,34,26,18,29,32)(21,30,37,39,36,35,38,40);(2,5)(6,7)(9,11,15,23)(10,16,12,20)(13,21,26,36)(14,28,17,22)(18,30,24,35)(19,33,25,31)(27,38,29,37)(32,40,34,39);(9,12,14,25,15,10,17,19)(11,16,22,31,23,20,28,33)(13,24,29,32,26,18,27,34)(21,30,38,40,36,35,37,39);(1,3,4,8)(2,6,5,7)(9,13)(10,18)(11,21)(12,24)(14,29)(15,26)(16,30)(17,27)(19,34)(20,35)(22,38)(23,36)(25,32)(28,37)(31,40)(33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282865);