#125204

order	512
nilpotency class	4
isoclinism	isoclinic class of #125092
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

325 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 92 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)(9,10,13,17,16,19,24,31,15,21,25,32,14,20,11,22)(12,18,26,30,29,34,38,33,28,36,39,40,27,35,23,37);(1,3)(2,5)(4,8)(6,7)(9,11,15,24)(10,17)(12,26,28,39)(13,16,25,14)(18,33)(19,22)(20,31)(21,32)(23,27,38,29)(30,34)(35,40)(36,37);(9,12,16,29,15,28,14,27)(10,18,19,34,21,36,20,35)(11,23,13,26,24,38,25,39)(17,30,31,33,32,40,22,37)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32400);