#146533

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
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

986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,d^b=dh,h^b=hi,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, 53729);