#244562

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

982 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:

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

Polycyclic presentation:

2^9|a^2=i,b^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, 151758);