#237927

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

872 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)(11,15)(12,16)(13,17)(14,18)(19,23)(20,22)(21,24)(25,26)(27,34)(28,32)(29,37)(30,33)(31,35)(36,40)(38,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,23)(18,22)(28,29)(31,39)(32,37)(35,38);(10,16)(11,19)(12,13)(17,18)(20,21)(23,24)(25,27)(26,31)(28,35)(29,38)(30,36)(32,40)(33,39)(34,37);(11,20)(15,22)(19,21)(23,24)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,40,39,34)

Polycyclic presentation:

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

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, 145123);