#146531

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

994 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)(7,16,18,12)(8,11)(9,23)(10,14)(13,20)(15,21)(17,29,34,24)(19,28,33,31)(22,25)(26,35)(27,32,30,36);(1,2,3,4)(6,11)(8,20)(9,21)(12,24)(13,15)(14,25)(16,29)(17,32)(19,33)(22,35)(23,26)(27,28)(30,31)(34,36);(5,7)(6,12)(8,17)(9,19)(10,18)(11,24)(13,27)(14,16)(15,28)(20,32)(21,33)(22,34)(23,31)(25,29)(26,30)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,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, 53727);