#380764

order	512
nilpotency class	3
isoclinism	isoclinic class of #29751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

990 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 290 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)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,18)(19,23)(20,30,33,24)(21,26)(22,25)(27,43,44,36)(28,37)(29,38)(31,47,46,39)(32,42,34,40)(35,41)(45,48);(2,6)(7,13)(8,14)(15,16)(17,19,21,29,22,28,35,45)(18,23,26,38,25,37,41,48)(20,31,32,44,33,46,34,27)(24,39,40,43,30,47,42,36);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,20)(18,24)(19,27)(21,34)(22,33)(23,36)(25,30)(26,42)(28,44)(29,46)(31,45)(32,35)(37,43)(38,47)(39,48)(40,41);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(19,28)(23,37)(27,44)(29,45)(31,46)(36,43)(38,48)(39,47)

Polycyclic presentation:

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

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, 287960);