#190191

order	512
nilpotency class	3
isoclinism	isoclinic class of #190116
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

835 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 835 conjugacy classes of subgroups — too many to draw. Only its 232 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,f^c=fh,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, 97387);