#201850

order	512
nilpotency class	3
isoclinism	isoclinic class of #201701
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1128 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)(9,13,11,15)(12,16)(17,18,21,26,22,25,27,40)(19,23,30,37,29,38,34,41)(20,31,33,42,32,24,35,39)(28,36,44,47,43,46,45,48);(1,3,5,10)(2,6)(4,11,12,9)(7,15)(8,14)(13,16)(17,19,27,30,22,29,21,34)(18,23,26,41,25,38,40,37)(20,28,33,45,32,43,35,44)(24,36,42,47,31,46,39,48);(17,20)(18,24)(19,28)(21,35)(22,32)(23,36)(25,31)(26,42)(27,33)(29,43)(30,45)(34,44)(37,48)(38,46)(39,40)(41,47);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(18,25)(19,29)(24,31)(26,40)(28,43)(30,34)(39,42)(44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,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, 109046);