#197409

order	512
nilpotency class	3
isoclinism	isoclinic class of #197354
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	#37558	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dhi,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, 104605);