#125438

order	512
nilpotency class	4
isoclinism	isoclinic class of #125438
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1558
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

246 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fg,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,g^a=gi,c^b=ch,d^b=dg,e^b=efg,f^b=fi,h^b=hi,e^c=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, 32634);