#133782

order	512
nilpotency class	4
isoclinism	isoclinic class of #133470
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	C₂×#117	C₂³
prime radical	#2539	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 586 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) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=eh,f^b=fg,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, 40978);