#202159

order	512
nilpotency class	3
isoclinism	isoclinic class of #201948
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	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

1013 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:

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

Polycyclic presentation:

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

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, 109355);