#219629

order	512
nilpotency class	3
isoclinism	isoclinic class of #219624
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

576 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=ef,d^2=ei,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dhi,f^b=fh,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, 126825);