#241254

order	512
nilpotency class	3
isoclinism	isoclinic class of #233734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

726 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148450);