#132628

order	512
nilpotency class	4
isoclinism	isoclinic class of #132541
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	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 522 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) = 128

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

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

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=egh,f^b=fgh,d^c=dghi,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39824);