#132424

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×#556 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 613 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)(19,28)(20,33,36,41)(21,26,37,25)(22,27)(23,30,39,31)(24,35)(29,38,45,48)(32,42)(34,40)(43,44)(46,47);(1,2)(3,7)(4,11)(5,8)(6,13)(9,14)(10,16)(12,15)(17,19,22,32)(18,23,27,39)(20,29,36,45)(21,30,37,31)(24,38,40,48)(25,28,26,42)(33,47,41,43)(34,44,35,46);(1,3,5,10)(2,7,8,16)(4,12,13,9)(6,14,11,15)(17,20,22,36)(18,24,27,40)(19,29,32,45)(21,35,37,34)(23,38,39,48)(25,41,26,33)(28,43,42,47)(30,46,31,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=h,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,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, 39620);