#96003

order	512
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

536 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=gi,e^2=gh,f^2=g,b^a=bd,d^a=dg,f^a=fghi,d^b=dh,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3199);