#194460

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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1019 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)(3,11,14,15)(4,13,16,8)(5,12)(6,9)(7,10)(17,18)(19,28,31,39)(20,32)(21,26)(22,36)(23,27)(24,30,40,35)(25,34)(29,44,45,38)(33,42)(37,41)(43,46,48,47);(1,3,5,13,7,14,6,8)(2,4,12,11,10,16,9,15)(17,19,21,35,23,31,22,30)(18,24,26,39,27,40,36,28)(20,29,34,47,33,45,37,46)(25,38,42,48,41,44,32,43);(1,4)(2,8)(3,12)(5,11)(6,15)(7,16)(9,14)(10,13)(17,20)(18,25)(19,29)(21,34)(22,37)(23,33)(24,38)(26,42)(27,41)(28,43)(30,46)(31,45)(32,36)(35,47)(39,48)(40,44);(1,5,7,6)(2,9,10,12)(3,13,14,8)(4,15,16,11)(20,33)(25,41)(29,45)(32,42)(34,37)(38,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=h,f^2=hi,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, 101656);