#195416

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1021 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,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, 102612);