#132412

order	512
nilpotency class	4
isoclinism	isoclinic class of #132245
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
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 ◁ C₂×#552 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

614 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,f^c=fg,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, 39608);