#194621

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
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

984 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101817);