#232734

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

1199 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1199 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:

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

Polycyclic presentation:

2^9|a^2=h,c^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, 139930);