#96553

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
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

540 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=g,e^2=i,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=cgi,d^b=dh

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, 3749);