#238969

order	512
nilpotency class	3
isoclinism	isoclinic class of #238962
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

813 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)(19,28)(20,33)(21,38)(22,26)(23,27)(24,31)(25,37)(29,45)(30,41)(32,40)(34,39)(35,43)(36,44)(42,47)(46,48);(1,3)(2,4)(5,12)(6,8)(7,13)(9,14)(10,15)(11,16)(17,19)(18,24)(20,34)(21,30)(22,31)(23,32)(25,42)(26,40)(27,41)(28,38)(29,35)(33,45)(36,47)(37,46)(39,44)(43,48);(1,4)(2,8)(3,9)(5,14)(6,15)(7,16)(10,13)(11,12)(17,20,21,36,23,37,22,35)(18,25,26,44,27,33,38,43)(19,29,31,46,32,47,30,34)(24,39,28,42,41,48,40,45);(1,5,7,6)(2,9,11,10)(3,12,13,8)(4,14,16,15)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,35,37,36)(24,40,41,28)(25,43,33,44)(29,46,47,34)(39,42,48,45)

Polycyclic presentation:

2^9|c^2=e,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=ghi,d^c=dh,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, 146165);