#123130

order	512
nilpotency class	4
isoclinism	isoclinic class of #123111
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

274 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 42 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)(3,11)(4,14)(5,9)(6,16)(7,12)(8,13)(10,15)(17,18,23,29)(19,30,33,45)(20,24,37,41)(21,28,22,27)(25,38,43,32)(26,36,40,34)(31,44,35,42)(39,47,46,48);(1,3,7,6)(2,8,12,14)(4,15,13,5)(9,11,10,16)(18,24)(20,34)(21,22)(25,39)(26,28)(27,40)(29,41)(30,44)(31,47)(32,38)(35,48)(36,37)(42,45)(43,46);(1,4,7,13)(2,9,12,10)(3,5,6,15)(8,16,14,11)(17,19,23,33)(18,25,29,43)(20,35,37,31)(21,38,22,32)(24,39,41,46)(26,44,40,42)(27,45,28,30)(34,48,36,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=fh,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ehi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30326);