#133196

order	512
nilpotency class	4
isoclinism	isoclinic class of #133175
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 463 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) = 128

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=g,d^2=hi,e^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,f^c=fhi,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, 40392);