#250511

order	512
nilpotency class	3
isoclinism	isoclinic class of #250455
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

1019 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)(3,10)(4,7)(5,9)(6,11)(8,14)(12,16)(13,15)(17,18,23,19)(20,24,32,27)(21,25,34,28)(22,29,37,26)(30,36,39,33)(31,41,44,38)(35,42,46,40)(43,45,48,47);(1,3)(2,6)(4,13)(5,12)(7,10)(8,16)(9,15)(11,14)(17,19,23,18)(20,30,32,39)(21,28,34,25)(22,26,37,29)(24,33,27,36)(31,43,44,48)(35,40,46,42)(38,45,41,47);(6,15)(10,16)(17,20,23,32)(18,24,19,27)(21,31,34,44)(22,36,37,33)(25,38,28,41)(26,30,29,39)(35,47,46,45)(40,43,42,48);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(17,21)(18,25)(19,28)(20,31)(22,35)(23,34)(24,38)(26,40)(27,41)(29,42)(30,43)(32,44)(33,45)(36,47)(37,46)(39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157707);