#496399

order	512
nilpotency class	4
isoclinism	isoclinic class of #480684
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

908 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 198 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,25)(21,26,33,24)(22,27)(28,35)(29,36,43,34)(30,37)(31,40)(32,39,44,38)(41,47)(42,46,48,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,30)(18,23,27,37)(20,28,31,41)(21,29,33,43)(24,34,26,36)(25,35,40,47)(32,42,44,48)(38,45,39,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(20,31)(21,33)(23,34)(25,39)(26,27)(28,41)(29,43)(35,46)(36,37)(38,40)(45,47);(2,8)(6,9)(7,16)(14,15)(17,20,22,31)(18,25,27,40)(19,28,30,41)(21,32,33,44)(23,35,37,47)(24,38,26,39)(29,42,43,48)(34,45,36,46)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,f^c=fi,g^c=ghi,g^f=gi

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, 403595);