#139415

order	512
nilpotency class	4
isoclinism	isoclinic class of #139356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 341 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) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=h,d^2=ghi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46611);