#197464

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	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1109 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:

(1,2,5,10,6,9,4,8)(3,11,12,7,14,15,13,16)(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)(2,7)(4,12)(5,13)(6,14)(8,15)(9,16)(10,11)(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);(1,4,6,5)(2,8,9,10)(3,12,14,13)(7,15,16,11)(17,20)(18,24)(19,28)(21,35)(22,32)(23,37)(25,40)(26,31)(27,41)(29,45)(30,43)(33,36)(34,39)(38,47)(42,48)(44,46);(2,9)(7,16)(8,10)(11,15)(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=i,e^2=hi,g^2=i,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, 104660);