#146561

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dhi,f^b=fi,h^b=hi,e^c=ei,f^c=fi

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, 53757);