#241113

order	512
nilpotency class	3
isoclinism	isoclinic class of #241105
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3192	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

847 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18)(19,29)(20,33)(21,26)(22,27)(23,28)(24,31)(25,36)(30,42)(32,40)(34,39)(35,44)(37,43)(38,41)(45,46)(47,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,34)(21,38)(22,31)(23,32)(25,42)(26,29)(27,40)(28,41)(30,37)(33,47)(35,46)(36,48)(39,44)(43,45);(4,11)(6,12)(7,14)(9,15)(17,20,21,35,23,37,22,36)(18,25,27,43,28,44,26,33)(19,30,31,46,32,34,38,48)(24,39,29,45,41,42,40,47);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,23,22)(18,26,28,27)(19,31,32,38)(20,35,37,36)(24,40,41,29)(25,33,44,43)(30,46,34,48)(39,47,42,45)

Polycyclic presentation:

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

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, 148309);