#219647

order	512
nilpotency class	3
isoclinism	isoclinic class of #219638
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	#532	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 501 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=ef,d^2=ehi,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,d^c=di,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, 126843);