#200576

order	512
nilpotency class	3
isoclinism	isoclinic class of #200418
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

1101 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)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,21,26,22,25,36,37)(19,23,29,40,28,39,34,43)(20,30,33,44,32,24,35,42)(27,38,45,47,31,41,46,48);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,19,21,34,22,28,36,29)(18,23,37,40,25,39,26,43)(20,31,35,46,32,27,33,45)(24,41,42,47,30,38,44,48);(3,11)(7,15)(10,16)(12,13)(17,20)(18,24)(19,27)(21,35)(22,32)(23,38)(25,30)(26,44)(28,31)(29,46)(33,36)(34,45)(37,42)(39,41)(40,48)(43,47);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(18,25)(23,39)(24,30)(26,37)(38,41)(40,43)(42,44)(47,48)

Polycyclic presentation:

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

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, 107772);