#241044

order	512
nilpotency class	3
isoclinism	isoclinic class of #241037
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	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

791 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=dhi,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, 148240);