#237520

order	512
nilpotency class	3
isoclinism	isoclinic class of #237505
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1215 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)(3,11,14,15)(4,8,5,9)(7,13,16,12)(17,18,23,28)(19,29,31,41)(20,33,35,42)(21,27,22,26)(24,36,39,32)(25,37,40,34)(30,38,44,48)(43,45,47,46);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,11,9,15)(17,19,23,31)(18,24,28,39)(20,30,35,44)(21,36,22,32)(25,38,40,48)(26,41,27,29)(33,47,42,43)(34,46,37,45);(3,12)(4,5)(7,11)(8,9)(13,14)(15,16)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,33)(27,42)(28,40)(29,43)(31,44)(32,45)(36,46)(39,48)(41,47);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(19,31)(24,39)(29,41)(30,44)(32,36)(38,48)(43,47)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144716);