#188934

order	512
nilpotency class	3
isoclinism	isoclinic class of #188824
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

815 conjugacy classes of subgroups; only the 276 normal subgroups are shown.

normal subgroups only

This group has 815 conjugacy classes of subgroups — too many to draw. Only its 276 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96130);