#194650

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

976 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,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, 101846);