#123188

order	512
nilpotency class	4
isoclinism	isoclinic class of #123171
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
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

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

normal subgroups only

This group has 264 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18,23,28)(19,25,20,31)(21,34,37,40)(22,39,36,27)(24,32,42,29)(26,45,46,38)(30,47,33,44)(35,43,48,41);(7,12)(9,15)(17,19,23,20)(18,24,28,42)(21,35,37,48)(22,33,36,30)(25,40,31,34)(26,44,46,47)(27,43,39,41)(29,45,32,38);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,20,23,19)(18,25)(21,32,37,29)(22,33,36,30)(24,40)(26,41)(27,44)(28,31)(34,42)(35,38,48,45)(39,47)(43,46)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,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,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=eh,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, 30384);