#240157

order	512
nilpotency class	3
isoclinism	isoclinic class of #10464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	#60559	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

918 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,37)(30,45)(31,40)(33,44)(35,39)(36,43)(38,42)(41,47)(46,48);(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,41)(26,29)(27,44)(28,40)(30,38)(34,45)(36,47)(37,46)(39,43)(42,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20,21,36,23,37,22,38)(18,25,27,43,28,34,26,42)(19,30,32,46,33,47,31,35)(24,39,29,41,40,48,44,45);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,37,38)(24,29,40,44)(25,42,34,43)(30,35,47,46)(39,41,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147353);