#232732

order	512
nilpotency class	3
isoclinism	isoclinic class of #232724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60940	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1249 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1249 conjugacy classes of subgroups — too many to draw. Only its 196 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)(3,10)(4,7)(5,9)(6,12)(8,14)(11,16)(13,15)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,45)(32,40)(33,39)(34,38)(35,42)(36,44)(37,43)(41,46)(47,48);(1,3,5,13)(2,6,9,16)(4,11,14,12)(7,10,8,15)(17,19,23,33)(18,24,28,40)(20,34,36,47)(21,32,22,31)(25,41,44,45)(26,29,27,39)(30,37,46,35)(38,43,48,42);(2,7)(3,11)(4,14)(8,9)(10,15)(12,13)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,42)(27,43)(28,44)(29,45)(31,34)(32,47)(33,46)(39,41)(40,48);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,42,44,43)(30,34,46,47)(38,41,48,45)

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139928);