#250626

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
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	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

998 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:

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

Polycyclic presentation:

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

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, 157822);