#239021

order	512
nilpotency class	3
isoclinism	isoclinic class of #239017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60967	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

833 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)(3,4)(5,9)(6,10)(7,11)(8,15)(12,14)(13,16)(17,18)(19,29)(20,34)(21,28)(22,27)(23,26)(24,31)(25,36)(30,45)(32,41)(33,40)(35,48)(37,43)(38,44)(39,47)(42,46);(1,3)(2,4)(5,12)(6,8)(7,13)(9,14)(10,15)(11,16)(17,19)(18,24)(20,35)(21,33)(22,31)(23,32)(25,42)(26,40)(27,41)(28,29)(30,38)(34,45)(36,47)(37,46)(39,44)(43,48);(1,4)(2,8)(3,9)(5,14)(6,15)(7,16)(10,13)(11,12)(17,20,21,37,23,36,22,38)(18,25,27,43,26,34,28,44)(19,30,31,47,32,46,33,35)(24,39,29,45,40,48,41,42);(1,5,7,6)(2,9,11,10)(3,12,13,8)(4,14,16,15)(18,26)(20,36)(24,40)(27,28)(29,41)(30,46)(35,47)(37,38)

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,f^c=fhi,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, 146217);