#237112

order	512
nilpotency class	3
isoclinism	isoclinic class of #237079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1169 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144308);