#196261

order	512
nilpotency class	3
isoclinism	isoclinic class of #196257
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	#2431	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1121 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ci,f^b=fh,e^c=eh,g^c=gh

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, 103457);