#133657

order	512
nilpotency class	4
isoclinism	isoclinic class of #133350
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	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#120	C₂×C₄
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 542 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,e^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,h^a=hi,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,f^c=fi,e^d=ei

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, 40853);