#199249

order	512
nilpotency class	3
isoclinism	isoclinic class of #199242
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1092 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)(4,7)(8,12)(9,13)(11,14)(15,16)(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);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19,23,31)(18,24)(20,29,36,45)(21,30,22,32)(25,37)(26,38)(27,40)(28,39)(33,43)(34,44,35,46)(41,47)(42,48);(1,3,5,10)(2,8,9,15)(4,11,7,14)(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);(3,10)(8,15)(11,14)(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=egh,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, 106445);