#123242

order	512
nilpotency class	4
isoclinism	isoclinic class of #123211
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	#1999	C₂²
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

317 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30438);