#232448

order	512
nilpotency class	3
isoclinism	isoclinic class of #232445
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1331 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18)(19,28)(20,33)(21,38)(22,27)(23,26)(24,30)(25,36)(29,46)(31,40)(32,41)(34,45)(35,43)(37,44)(39,47)(42,48);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,42)(26,40)(27,41)(28,38)(29,35)(33,39)(36,47)(37,48)(43,45)(44,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20,23,37)(18,25,26,43)(19,29,32,47)(21,35,22,36)(24,39,40,46)(27,44,38,33)(28,45,41,42)(30,34,31,48);(18,26)(24,40)(25,43)(27,38)(28,41)(33,44)(39,46)(42,45)

Polycyclic presentation:

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

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, 139644);