#133763

order	512
nilpotency class	4
isoclinism	isoclinic class of #133290
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	#35394	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 840 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,11,14,15)(4,9,5,8)(6,10)(7,13)(12,16)(17,18,21,24,22,26,19,27)(20,23,28,32,29,25,30,31);(1,3)(2,7)(4,13)(5,12)(6,14)(8,11)(9,15)(10,16)(18,23)(19,21)(20,29)(24,32)(25,26)(27,31);(1,4)(2,8)(3,12)(5,6)(7,15)(9,10)(11,16)(13,14)(17,19)(18,24)(20,28)(21,22)(23,31)(25,32)(26,27)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40959);