#200311

order	512
nilpotency class	3
isoclinism	isoclinic class of #200158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1048 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fi,g^c=gh

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, 107507);