#127364

order	512
nilpotency class	4
isoclinism	isoclinic class of #127265
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

449 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 170 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,29)(20,34)(21,25)(22,39)(23,28)(24,31)(26,37)(27,36)(30,43)(32,42)(33,48)(35,40)(38,44)(41,47)(45,46);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,21,32,22,33,37,31,23,30,38,46,36,47,20,35)(18,24,26,41,27,42,25,43,28,40,34,48,39,45,44,29);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,30)(24,40)(29,43)(31,35)(32,46)(33,47)(41,48)(42,45)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=ghi,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=ci,d^b=di,e^b=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34560);