#198536

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	#60946	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1054 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:

(1,2,6,9)(3,5,8,15)(4,11,14,10)(7,12,16,13)(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);(1,3,5,9,6,8,15,2)(4,10,12,16,14,11,13,7)(17,19,22,29)(18,23)(20,27,32,40)(21,28,34,42)(24,35)(25,36)(26,37)(30,43)(31,39,45,48)(33,41)(38,46)(44,47);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,16)(13,15)(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);(1,5,6,15)(2,8,9,3)(4,12,14,13)(7,11,16,10)(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=egh,d^2=h,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, 105732);