#197247

order	512
nilpotency class	3
isoclinism	isoclinic class of #196993
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60559	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1143 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104443);