#232166

order	512
nilpotency class	3
isoclinism	isoclinic class of #232153
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19750	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

443 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=efi,d^2=eh,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cef,d^b=de,e^b=ei,f^b=fhi,g^b=ghi,g^c=gh,g^d=gi

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, 139362);