#201269

order	512
nilpotency class	3
isoclinism	isoclinic class of #201121
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1075 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)(3,9)(4,12,13,7)(5,11)(6,8)(10,14,16,15)(17,18,22,26,23,27,21,33)(19,24,29,38,30,39,35,37)(20,25,31,42,32,41,34,40)(28,36,45,46,44,47,43,48);(1,3,9,2,6,11,5,8)(4,10,14,7,13,16,15,12)(17,19,21,29,23,30,22,35)(18,24,26,37,27,39,33,38)(20,28,34,45,32,44,31,43)(25,36,42,48,41,47,40,46);(1,4)(2,7)(3,10)(5,15)(6,13)(8,12)(9,14)(11,16)(17,20)(18,25)(19,28)(21,34)(22,31)(23,32)(24,36)(26,42)(27,41)(29,45)(30,44)(33,40)(35,43)(37,48)(38,46)(39,47);(4,13)(7,12)(10,16)(14,15)(17,21,23,22)(18,26,27,33)(19,29,30,35)(20,31,32,34)(24,37,39,38)(25,40,41,42)(28,43,44,45)(36,46,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108465);