#198700

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	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1005 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=h,e^2=hi,f^2=hi,g^2=i,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, 105896);