#195375

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1055 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:

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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102571);