#198537

order	512
nilpotency class	3
isoclinism	isoclinic class of #198534
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	C₂³×C₄	C₂⁴
prime radical	#60942	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1003 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)(3,9)(4,12)(8,14)(10,11)(13,16)(17,18)(19,23,29,37)(20,30,32,24)(21,33)(22,26)(25,34)(27,35)(28,41,42,36)(31,38,45,44)(39,47)(40,43)(46,48);(1,2,4,6,5,8,12,14)(3,7,10,13,11,15,9,16)(17,19)(18,23,26,37)(20,27)(21,28)(22,29)(24,35,30,43)(25,36,33,41)(31,39)(32,40)(34,42)(38,46,44,47)(45,48);(1,3)(2,7)(4,10)(5,11)(6,13)(8,15)(9,12)(14,16)(17,20)(18,24)(19,27)(21,31)(22,32)(23,35)(25,38)(26,30)(28,39)(29,40)(33,44)(34,45)(36,46)(37,43)(41,47)(42,48);(2,8)(6,14)(7,15)(13,16)(17,21,22,34)(18,25,26,33)(19,28,29,42)(20,31,32,45)(23,36,37,41)(24,38,30,44)(27,39,40,48)(35,46,43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,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, 105733);