#199248

order	512
nilpotency class	3
isoclinism	isoclinic class of #199242
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	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

990 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:

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

Polycyclic presentation:

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

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, 106444);