#240729

order	512
nilpotency class	3
isoclinism	isoclinic class of #240721
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

798 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)(3,7)(4,9)(5,8)(6,10)(11,14)(12,16)(13,15)(17,18,23,27)(19,28,32,42)(20,33,37,25)(21,26,22,38)(24,30,40,31)(29,43,34,48)(35,47,36,41)(39,44,46,45);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,39)(26,42)(27,40)(28,38)(29,37)(33,46)(35,44)(36,45)(41,48)(43,47);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,20,22,36,23,37,21,35)(18,25,26,41,27,33,38,47)(19,29,31,45,32,34,30,44)(24,39,28,43,40,46,42,48);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,35,37,36)(24,28,40,42)(25,41,33,47)(29,44,34,45)(39,43,46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147925);