#244574

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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,7,10)(3,11,14,15)(4,12,16,13)(5,8,6,9)(17,18,23,28)(19,29,33,40)(20,34,36,43)(21,27,22,26)(24,32,41,31)(25,37,42,35)(30,39,46,38)(44,47,48,45);(2,8)(3,12)(5,6)(9,10)(11,15)(13,14)(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);(1,3)(2,4)(5,13)(6,12)(7,14)(8,15)(9,11)(10,16)(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)(2,3)(5,11)(6,15)(7,16)(8,12)(9,13)(10,14)(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=h,b^2=hi,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, 151770);