#123185

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

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

normal subgroups only

This group has 256 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)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(17,18,22,27)(19,28,30,24)(20,23,35,38)(21,36,34,26)(25,42,39,32)(29,43,33,37)(31,41,47,45)(40,46,44,48);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,14,8,15)(18,23)(20,32)(21,34)(24,37)(25,26)(27,38)(28,43)(29,46)(31,47)(33,48)(35,42)(36,39)(40,41)(44,45);(2,7,9,8)(6,14,16,15)(17,19,22,30)(18,24)(20,33,35,29)(21,31,34,47)(23,37)(25,40)(26,41)(27,28)(32,48,42,46)(36,45)(38,43)(39,44)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,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, 30381);