#232168

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	#532	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 423 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)(3,12,14,16)(4,15,13,8)(5,10)(6,9)(7,11)(17,18)(19,27,30,23)(20,31,35,46)(21,25)(22,26)(24,34,40,44)(28,39)(29,41,36,38)(32,37)(33,42)(43,47)(45,48);(1,3)(2,8)(4,6)(5,13)(7,14)(9,12)(10,16)(11,15)(17,19,22,30)(18,23,26,27)(20,32,35,47)(21,36,33,29)(24,39,40,48)(25,41,42,38)(28,44,45,34)(31,43,46,37);(2,9,11,10)(8,16,15,12)(17,20,33,44,22,35,21,34)(18,24,25,31,26,40,42,46)(19,28,36,47,30,45,29,32)(23,37,38,48,27,43,41,39);(1,4,6,3,7,13,5,14)(2,8,9,16,11,15,10,12)(18,25,26,42)(23,38,27,41)(24,31,40,46)(37,48,43,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=efh,d^2=ei,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, 139364);