#195055

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	#34636	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,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, 102251);