#129389

order	512
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

586 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 122 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)(17,18,22,29,23,30,38,26,24,28,37,45,21,27,20,36)(19,31,34,48,33,43,32,47,35,46,40,42,39,25,41,44);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,25)(20,32)(21,39)(22,40)(23,33)(24,35)(26,42)(27,46)(28,43)(29,47)(30,31)(34,37)(36,48)(38,41)(44,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,23,22,24,38,21,37)(18,26,30,45,28,36,27,29)(19,32,33,40,35,41,39,34)(25,42,31,44,43,48,46,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=di,f^b=fgi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36585);