#132500

order	512
nilpotency class	4
isoclinism	isoclinic class of #132317
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2353 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,e^c=ei,f^c=fgi,f^d=fi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39696);