#132046

order	512
nilpotency class	4
isoclinism	isoclinic class of #131970
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

634 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

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

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, 39242);