#122756

order	512
nilpotency class	4
isoclinism	isoclinic class of #122714
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

298 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=gh,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cfi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=dhi,e^d=ehi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29952);