#196302

order	512
nilpotency class	3
isoclinism	isoclinic class of #196299
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1081 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:

(1,2,6,9,5,8,14,16)(3,4,11,13,10,12,7,15)(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);(1,3,6,11,5,10,14,7)(2,4,9,13,8,12,16,15)(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);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,20,23,35)(18,25,27,33)(19,29,32,45)(21,34,22,37)(24,38,39,47)(26,40,36,41)(28,42,43,48)(30,44,31,46);(2,8)(4,12)(9,16)(13,15)(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,c^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,f^b=fh,d^c=di,e^c=eh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103498);