#148843

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56039);