#129632

order	512
nilpotency class	4
isoclinism	isoclinic class of #129627
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂³	#379
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

504 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=chi,d^b=di,e^b=eg,f^b=fgi,d^c=dgi,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36828);