#244051

order	512
nilpotency class	3
isoclinism	isoclinic class of #233469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151247);