#133205

order	512
nilpotency class	4
isoclinism	isoclinic class of #133182
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2313 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

476 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,10,12,7)(4,8)(5,9)(6,11,15,13)(14,16)(17,18,21,26,22,25,33,39)(19,23,29,35,28,36,45,40)(20,24,31,37,30,38,32,41)(27,42,43,48,44,34,46,47);(1,3)(2,6)(4,13)(5,12)(7,8)(9,15)(10,16)(11,14)(17,19)(18,23,25,36)(20,27)(21,29)(22,28)(24,34,38,42)(26,40,39,35)(30,44)(31,43)(32,46)(33,45)(37,48,41,47);(1,3)(2,7)(4,11)(5,12)(6,8)(9,10)(13,14)(15,16)(17,20,22,30)(18,24)(19,27,28,44)(21,32,33,31)(23,34)(25,38)(26,41)(29,46,45,43)(35,48)(36,42)(37,39)(40,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gi,d^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,e^c=ei,f^c=fh,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40401);