#197345

order	512
nilpotency class	3
isoclinism	isoclinic class of #197339
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	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1121 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18)(19,27)(20,31,32,24)(21,34)(22,26)(23,29)(25,36)(28,41,43,48)(30,42)(33,39,35,40)(37,44,47,45)(38,46);(1,3,5,14,6,12,4,11)(2,7,8,15,9,16,13,10)(17,19,21,29,22,30,36,46)(18,23,25,27,26,38,34,42)(20,28,35,45,32,43,33,44)(24,37,40,41,31,47,39,48);(17,20,22,32)(18,24,26,31)(19,28,30,43)(21,35,36,33)(23,37,38,47)(25,40,34,39)(27,41,42,48)(29,45,46,44);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(20,32)(24,31)(28,43)(33,35)(37,47)(39,40)(41,48)(44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,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, 104541);