#330641

order	512
nilpotency class	3
isoclinism	isoclinic class of #329647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

550 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=i,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

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, 237837);