#132413

order	512
nilpotency class	4
isoclinism	isoclinic class of #132253
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#2631	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#552 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 638 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) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,e^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,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, 39609);