#125067

order	512
nilpotency class	4
isoclinism	isoclinic class of #125064
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

352 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28,31,46)(20,25)(21,34,36,44)(22,38)(23,27)(24,30)(26,33,43,37)(29,45,32,42)(35,39)(40,47)(41,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,23,31)(18,24,27,41)(20,29,22,32)(21,35,36,47)(25,39,38,40)(26,42,43,45)(28,44,46,34)(30,33,48,37);(3,9)(6,12)(10,15)(13,16)(17,20,23,22)(18,25,27,38)(19,29,31,32)(21,33,36,37)(24,39,41,40)(26,34,43,44)(28,45,46,42)(30,35,48,47)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,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, 32263);