#198805

order	512
nilpotency class	3
isoclinism	isoclinic class of #198798
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	#60965	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1096 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)(3,6,10,14)(4,7)(5,8)(9,12)(11,13,16,15)(17,18)(19,28)(20,33,35,25)(21,26)(22,36)(23,27)(24,31)(29,42,45,48)(30,39)(32,43)(34,41,37,40)(38,44,47,46);(1,3)(2,6,8,14)(4,11)(5,10)(7,13,9,15)(12,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,36,43)(20,29,37,46,35,45,34,44)(25,38,40,42,33,47,41,48);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,40)(27,33)(28,42)(30,44)(31,46)(32,45)(36,41)(39,47)(43,48);(2,8)(6,14)(7,9)(13,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,35,37)(24,28,39,43)(25,40,33,41)(29,44,45,46)(38,42,47,48)

Polycyclic presentation:

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

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, 106001);