#125463

order	512
nilpotency class	4
isoclinism	isoclinic class of #125409
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

290 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=hi,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efg,f^b=fi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32659);