#175201

order	512
nilpotency class	3
isoclinism	isoclinic class of #174829
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

678 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 82397);