#244565

order	512
nilpotency class	3
isoclinism	isoclinic class of #244547
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62598	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

964 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)(3,12)(4,14)(5,8)(6,10)(7,11)(9,15)(13,16)(17,18,23,28)(19,29,32,38)(20,33,35,42)(21,24,22,27)(25,37,40,31)(26,36,41,34)(30,39,46,48)(43,45,47,44);(2,8)(3,13)(5,6)(9,14)(10,11)(12,15)(18,24)(21,22)(25,29)(26,33)(27,28)(31,37)(34,36)(38,40)(39,43)(41,42)(44,45)(47,48);(1,3,7,9)(2,4,11,16)(5,14,6,13)(8,15,10,12)(17,19,23,32)(18,25,28,40)(20,30,35,46)(21,31,22,37)(24,38,27,29)(26,39,41,48)(33,47,42,43)(34,44,36,45);(1,4)(2,9)(3,11)(5,12)(6,15)(7,16)(8,14)(10,13)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,33)(25,39)(27,42)(28,41)(29,43)(31,45)(32,46)(37,44)(38,47)(40,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fh,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151761);