#124693

order	512
nilpotency class	4
isoclinism	isoclinic class of #124651
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

371 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31889);