#201904

order	512
nilpotency class	3
isoclinism	isoclinic class of #201717
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2617	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1053 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109100);