#123025

order	512
nilpotency class	4
isoclinism	isoclinic class of #123011
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×C₂²⋊C₈	C₂³
prime radical	C₂²×#41	D₄
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

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

normal subgroups only

This group has 364 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)(4,7)(5,8)(6,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,30,33,44)(20,24,35,40)(21,28,22,27)(25,36,42,32)(26,37,39,34)(31,43,48,41)(38,46,45,47);(1,3,5,11)(2,6,8,13)(4,10,12,16)(7,9,14,15)(18,24)(20,34)(21,22)(25,38)(26,28)(27,39)(29,40)(30,43)(31,46)(32,36)(35,37)(41,44)(42,45)(47,48);(4,12)(10,16)(17,19)(18,25)(20,31)(21,36)(22,32)(23,33)(24,38)(26,43)(27,44)(28,30)(29,42)(34,46)(35,48)(37,47)(39,41)(40,45)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,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, 30221);